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Theorem nssne1 3327
 Description: Two classes are different if they don't include the same class. (Contributed by NM, 23-Apr-2015.)
Assertion
Ref Expression
nssne1 ((A B ¬ A C) → BC)

Proof of Theorem nssne1
StepHypRef Expression
1 sseq2 3293 . . . 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   ≠ wne 2516   ⊆ wss 3257 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-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259 This theorem is referenced by: (None)
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