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 | |- E. x -. x = y |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | exexneq | |- E. z E. 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 -> E. x -. x = y ) |
| 7 | 3 6 | syl6 | |- ( w = y -> ( -. z = w -> E. 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 -> E. x -. x = y ) |
| 11 | 10 | a1d | |- ( -. w = y -> ( -. z = w -> E. x -. x = y ) ) |
| 12 | 7 11 | pm2.61i | |- ( -. z = w -> E. x -. x = y ) |
| 13 | 12 | exlimivv | |- ( E. z E. w -. z = w -> E. x -. x = y ) |
| 14 | 1 13 | ax-mp | |- E. x -. x = y |