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Theorem nelne1 2430
Description: Two classes are different if they don't contain the same element. (Contributed by NM, 3-Feb-2012.)
Assertion
Ref Expression
nelne1 ((𝐴𝐵 ∧ ¬ 𝐴𝐶) → 𝐵𝐶)

Proof of Theorem nelne1
StepHypRef Expression
1 eleq2 2234 . . . 4 (𝐵 = 𝐶 → (𝐴𝐵𝐴𝐶))
21biimpcd 158 . . 3 (𝐴𝐵 → (𝐵 = 𝐶𝐴𝐶))
32necon3bd 2383 . 2 (𝐴𝐵 → (¬ 𝐴𝐶𝐵𝐶))
43imp 123 1 ((𝐴𝐵 ∧ ¬ 𝐴𝐶) → 𝐵𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103   = wceq 1348  wcel 2141  wne 2340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-cleq 2163  df-clel 2166  df-ne 2341
This theorem is referenced by:  elnelne1  2444  difsnb  3721
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