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Theorem nelne1 2605
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 2414 . . . 4 (B = C → (A BA C))
21biimpcd 215 . . 3 (A B → (B = CA C))
32necon3bd 2553 . 2 (A B → (¬ A CBC))
43imp 418 1 ((A B ¬ A C) → BC)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   wa 358   = wceq 1642   wcel 1710  wne 2516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-clel 2349  df-ne 2518
This theorem is referenced by:  difsnb  3850
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