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Theorem dtruarb 3933
Description: At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). This theorem asserts the existence of two sets which do not equal each other; compare with dtruex 4237 in which we are given a set y and go from there to a set x which is not equal to it. (Contributed by Jim Kingdon, 2-Sep-2018.)
Assertion
Ref Expression
dtruarb xy ¬ x = y
Distinct variable group:   x,y

Proof of Theorem dtruarb
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 el 3922 . . 3 x z x
2 ax-nul 3874 . . . 4 yz ¬ z y
3 sp 1398 . . . 4 (z ¬ z y → ¬ z y)
42, 3eximii 1490 . . 3 y ¬ z y
5 eeanv 1804 . . 3 (xy(z x ¬ z y) ↔ (x z x y ¬ z y))
61, 4, 5mpbir2an 848 . 2 xy(z x ¬ z y)
7 nelneq2 2136 . . 3 ((z x ¬ z y) → ¬ x = y)
872eximi 1489 . 2 (xy(z x ¬ z y) → xy ¬ x = y)
96, 8ax-mp 7 1 xy ¬ x = y
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   wa 97  wal 1240  wex 1378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019  ax-nul 3874  ax-pow 3918
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-cleq 2030  df-clel 2033
This theorem is referenced by: (None)
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