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Theorem disjel 3597
 Description: A set can't belong to both members of disjoint classes. (Contributed by NM, 28-Feb-2015.)
Assertion
Ref Expression
disjel (((AB) = C A) → ¬ C B)

Proof of Theorem disjel
StepHypRef Expression
1 disj3 3595 . . 3 ((AB) = A = (A B))
2 eleq2 2414 . . . 4 (A = (A B) → (C AC (A B)))
3 eldifn 3389 . . . 4 (C (A B) → ¬ C B)
42, 3syl6bi 219 . . 3 (A = (A B) → (C A → ¬ C B))
51, 4sylbi 187 . 2 ((AB) = → (C A → ¬ C B))
65imp 418 1 (((AB) = C A) → ¬ C B)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ∖ cdif 3206   ∩ cin 3208  ∅c0 3550 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551 This theorem is referenced by: (None)
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