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Theorem nelne1 2273
Description: Two classes are different if they don't contain the same element. (Contributed by NM, 3-Feb-2012.)
Assertion
Ref Expression
nelne1 ((A B ¬ A 𝐶) → B𝐶)

Proof of Theorem nelne1
StepHypRef Expression
1 eleq2 2083 . . . 4 (B = 𝐶 → (A BA 𝐶))
21biimpcd 148 . . 3 (A B → (B = 𝐶A 𝐶))
32necon3bd 2226 . 2 (A B → (¬ A 𝐶B𝐶))
43imp 115 1 ((A B ¬ A 𝐶) → B𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   = wceq 1228   wcel 1374  wne 2186
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-5 1316  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-4 1381  ax-17 1400  ax-ial 1409  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-cleq 2015  df-clel 2018  df-ne 2188
This theorem is referenced by:  difsnb  3480
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