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Theorem nssne1 3001
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 2967 . . . 4 (𝐵 = 𝐶 → (𝐴𝐵𝐴𝐶))
21biimpcd 148 . . 3 (𝐴𝐵 → (𝐵 = 𝐶𝐴𝐶))
32necon3bd 2248 . 2 (𝐴𝐵 → (¬ 𝐴𝐶𝐵𝐶))
43imp 115 1 ((𝐴𝐵 ∧ ¬ 𝐴𝐶) → 𝐵𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 97   = wceq 1243  wne 2204  wss 2917
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-ne 2206  df-in 2924  df-ss 2931
This theorem is referenced by: (None)
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