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Theorem dtru 4238
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 4237. (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 4237 . 2 x ¬ x = y
2 exnalim 1534 . 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 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-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-setind 4220
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-v 2553  df-dif 2914  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373
This theorem is referenced by:  oprabidlem  5479
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