JFrame.setVisible(true/false);

import javax.swing.*;
import java.awt.*;
import java.util.*;
 
public class MainPage extends JFrame {
	private MainPagePanel mpp;
 
	public MainPage() {
		super("Test");
		setSize(180,100);
		setDefaultCloseOperation(JFrame.HIDE_ON_CLOSE);
		mpp = new MainPagePanel();
		Container pane = getContentPane();
		pane.add(mpp);
		setContentPane(mpp);
		setResizable(false);
		setVisible(true);
	}
 
	public static void main(String[] args) {
 
		MainPage mp = new MainPage();
	}
}

import java.awt.*;
import java.awt.event.MouseEvent;
import java.awt.event.MouseListener;
import javax.swing.*;
 
public class MainPagePanel extends JPanel {
 
	ImageIcon image;
	JLabel display;
 
	public MainPagePanel() {
		Toolkit kit = Toolkit.getDefaultToolkit();
 
		image = new ImageIcon("menu1.jpg");
		display = new MapLabel(image, "Display");
        display.setBounds(10, 20, image.getIconWidth(), image.getIconHeight());
 
        setPreferredSize(new Dimension(457, 540));
        setLayout(null);
        add(display);
 
	}
 
	class MapLabel extends JLabel implements MouseListener {
		Icon icon;
        String name;
        MainPage mp; // Declaring MainPage mp = new MainPage(); here will result to error
 
        public MapLabel(Icon icon, String name) {
		    this.icon = icon;
		    this.name = name;
 
		    setIcon(icon);
		    setToolTipText(name);
            addMouseListener(this);
		}
 
   	public void mouseClicked(MouseEvent e) {
		String image = name;
 
		if(image.equals("Display")) {
			// How can I close the MainPage program if I click the display image label before PageOne will appear?
			// setVisibility won't work.
			PageOne po = new PageOne();
			mp.setVisible(false); // disposing PageOne
			mp.dispose(); //disposing PageOne
			mp = new MainPage();
        }
 
	}
	public void mouseEntered(MouseEvent e) {
            setCursor(Cursor.getPredefinedCursor(Cursor.HAND_CURSOR));
    }
    public void mouseExited(MouseEvent e) {
            setCursor(null);
    }
    public void mousePressed(MouseEvent e) {}
	public void mouseReleased(MouseEvent e) {}
	}
}

MainPage mp; 

public MapLabel(Icon icon, String name, MainPage mp) {

   this.mp = mp;
}

// AT THE LISTENER:

......
PageOne po = new PageOne();
mp.setVisible(false); // disposing PageOne

// mp = new MainPage(); // you don't need that. It will create a new MainPage and overwrite the existing one.
// you have already created a MainPage, don't create another

MainPage mp;

public MainPagePanel(MainPage mp) {
		Toolkit kit = Toolkit.getDefaultToolkit();

		image = new ImageIcon("menu1.jpg");
                this.mp = mp;
display = new MapLabel(image, "Display", mp);

public MainPage() {
		super("Test");
		setSize(180,100);
		setDefaultCloseOperation(JFrame.HIDE_ON_CLOSE);

// "this" is a MainPage. This MainPage that you just created and has all your Panels and labels
		mpp = new MainPagePanel(this);

		Container pane = getContentPane();
		pane.add(mpp);
		setContentPane(mpp);
		setResizable(false);
		setVisible(true);
	}

class PageOne {
  MainPage mp;
 
public PageOne(MainPage mp) {
 this.mp = mp;
}
 
....
...
// in some method:
{
   this.setVisble(false); //hiding PageOne
   mp.setVisible(true);
}
}

mp.setVisible(true);
this.dispose();

addMouseListener(this);

this.addMouseListener(this);