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Theorem nssne2 3187
Description: Two classes are different if they are not subclasses of the same class. (Contributed by NM, 23-Apr-2015.)
Assertion
Ref Expression
nssne2 ((𝐴𝐶 ∧ ¬ 𝐵𝐶) → 𝐴𝐵)

Proof of Theorem nssne2
StepHypRef Expression
1 sseq1 3151 . . . 4 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
21biimpcd 158 . . 3 (𝐴𝐶 → (𝐴 = 𝐵𝐵𝐶))
32necon3bd 2370 . 2 (𝐴𝐶 → (¬ 𝐵𝐶𝐴𝐵))
43imp 123 1 ((𝐴𝐶 ∧ ¬ 𝐵𝐶) → 𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103   = wceq 1335  wne 2327  wss 3102
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 604  ax-in2 605  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-ne 2328  df-in 3108  df-ss 3115
This theorem is referenced by: (None)
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