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Theorem dtru 4222
 Description: At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). If we assumed the law of the excluded middle this would be equivalent to dtruex 4221. (Contributed by Jim Kingdon, 29-Dec-2018.)
Assertion
Ref Expression
dtru ¬ x x = y
Distinct variable group:   x,y

Proof of Theorem dtru
StepHypRef Expression
1 dtruex 4221 . 2 x ¬ x = y
2 exnalim 1519 . 2 (x ¬ x = y → ¬ x x = y)
31, 2ax-mp 7 1 ¬ x x = y
 Colors of variables: wff set class Syntax hints:  ¬ wn 3  ∀wal 1226  ∃wex 1362 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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-setind 4204 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-v 2537  df-dif 2897  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356 This theorem is referenced by:  oprabidlem  5460
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