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Theorem disj 3591
 Description: Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 17-Feb-2004.)
Assertion
Ref Expression
disj ((AB) = x A ¬ x B)
Distinct variable groups:   x,A   x,B

Proof of Theorem disj
StepHypRef Expression
1 elin 3219 . . . . 5 (x (AB) ↔ (x A x B))
2 df-an 360 . . . . 5 ((x A x B) ↔ ¬ (x A → ¬ x B))
31, 2bitr2i 241 . . . 4 (¬ (x A → ¬ x B) ↔ x (AB))
43con1bii 321 . . 3 x (AB) ↔ (x A → ¬ x B))
54albii 1566 . 2 (x ¬ x (AB) ↔ x(x A → ¬ x B))
6 eq0 3564 . 2 ((AB) = x ¬ x (AB))
7 df-ral 2619 . 2 (x A ¬ x Bx(x A → ¬ x B))
85, 6, 73bitr4i 268 1 ((AB) = x A ¬ x B)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  ∀wral 2614   ∩ cin 3208  ∅c0 3550 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-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:  disjr  3592  disj1  3593  disjne  3596  disj5  3890  pw10  4161  pw1disj  4167  phidisjnn  4615  fvun1  5379  xpnedisj  5513  disjex  5823
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