Metamath Proof Explorer


Theorem exneq

Description: Given any set (the " y " in the statement), there exists a set not equal to it.

The same statement without disjoint variable condition is false, since we do not have E. x -. x = x . This theorem is proved directly from set theory axioms (no class definitions) and does not depend on ax-ext , ax-sep , or ax-pow nor auxiliary logical axiom schemes ax-10 to ax-13 . See dtruALT for a shorter proof using more axioms, and dtruALT2 for a proof using ax-pow instead of ax-pr . (Contributed by NM, 7-Nov-2006) Avoid ax-13 . (Revised by BJ, 31-May-2019) Avoid ax-8 . (Revised by SN, 21-Sep-2023) Avoid ax-12 . (Revised by Rohan Ridenour, 9-Oct-2024) Use ax-pr instead of ax-pow . (Revised by BTernaryTau, 3-Dec-2024) Extract this result from the proof of dtru . (Revised by BJ, 2-Jan-2025)

Ref Expression
Assertion exneq x ¬ x = y

Proof

Step Hyp Ref Expression
1 exexneq z w ¬ z = w
2 equeuclr w = y z = y z = w
3 2 con3d w = y ¬ z = w ¬ z = y
4 ax7v1 x = z x = y z = y
5 4 con3d x = z ¬ z = y ¬ x = y
6 5 spimevw ¬ z = y x ¬ x = y
7 3 6 syl6 w = y ¬ z = w x ¬ x = y
8 ax7v1 x = w x = y w = y
9 8 con3d x = w ¬ w = y ¬ x = y
10 9 spimevw ¬ w = y x ¬ x = y
11 10 a1d ¬ w = y ¬ z = w x ¬ x = y
12 7 11 pm2.61i ¬ z = w x ¬ x = y
13 12 exlimivv z w ¬ z = w x ¬ x = y
14 1 13 ax-mp x ¬ x = y