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Theorem nelneq 2135
Description: A way of showing two classes are not equal. (Contributed by NM, 1-Apr-1997.)
Assertion
Ref Expression
nelneq ((A 𝐶 ¬ B 𝐶) → ¬ A = B)

Proof of Theorem nelneq
StepHypRef Expression
1 eleq1 2097 . . 3 (A = B → (A 𝐶B 𝐶))
21biimpcd 148 . 2 (A 𝐶 → (A = BB 𝐶))
32con3and 563 1 ((A 𝐶 ¬ B 𝐶) → ¬ A = B)
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: (None)
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