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Theorem nelneq2 2136
Description: A way of showing two classes are not equal. (Contributed by NM, 12-Jan-2002.)
Assertion
Ref Expression
nelneq2 ((A B ¬ A 𝐶) → ¬ B = 𝐶)

Proof of Theorem nelneq2
StepHypRef Expression
1 eleq2 2098 . . 3 (B = 𝐶 → (A BA 𝐶))
21biimpcd 148 . 2 (A B → (B = 𝐶A 𝐶))
32con3and 563 1 ((A B ¬ A 𝐶) → ¬ B = 𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   = wceq 1242   wcel 1390
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-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-cleq 2030  df-clel 2033
This theorem is referenced by:  ssnelpss  3283  dtruarb  3933  fzneuz  8693
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