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Theorem nssne1 3155
 Description: Two classes are different if they don't include the same class. (Contributed by NM, 23-Apr-2015.)
Assertion
Ref Expression
nssne1 ((𝐴𝐵 ∧ ¬ 𝐴𝐶) → 𝐵𝐶)

Proof of Theorem nssne1
StepHypRef Expression
1 sseq2 3121 . . . 4 (𝐵 = 𝐶 → (𝐴𝐵𝐴𝐶))
21biimpcd 158 . . 3 (𝐴𝐵 → (𝐵 = 𝐶𝐴𝐶))
32necon3bd 2351 . 2 (𝐴𝐵 → (¬ 𝐴𝐶𝐵𝐶))
43imp 123 1 ((𝐴𝐵 ∧ ¬ 𝐴𝐶) → 𝐵𝐶)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   = wceq 1331   ≠ wne 2308   ⊆ wss 3071 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 603  ax-in2 604  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-ne 2309  df-in 3077  df-ss 3084 This theorem is referenced by: (None)
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