New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  disjr GIF version

Theorem disjr 3592
 Description: Two ways of saying that two classes are disjoint. (Contributed by Jeff Madsen, 19-Jun-2011.)
Assertion
Ref Expression
disjr ((AB) = x B ¬ x A)
Distinct variable groups:   x,A   x,B

Proof of Theorem disjr
StepHypRef Expression
1 incom 3448 . . 3 (AB) = (BA)
21eqeq1i 2360 . 2 ((AB) = ↔ (BA) = )
3 disj 3591 . 2 ((BA) = x B ¬ x A)
42, 3bitri 240 1 ((AB) = x B ¬ x A)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 176   = wceq 1642   ∈ wcel 1710  ∀wral 2614   ∩ 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:  nnadjoinpw  4521  sfinltfin  4535
 Copyright terms: Public domain W3C validator