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Theorem disjr 3970
 Description: Two ways of saying that two classes are disjoint. (Contributed by Jeff Madsen, 19-Jun-2011.)
Assertion
Ref Expression
disjr ((𝐴𝐵) = ∅ ↔ ∀𝑥𝐵 ¬ 𝑥𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem disjr
StepHypRef Expression
1 incom 3767 . . 3 (𝐴𝐵) = (𝐵𝐴)
21eqeq1i 2615 . 2 ((𝐴𝐵) = ∅ ↔ (𝐵𝐴) = ∅)
3 disj 3969 . 2 ((𝐵𝐴) = ∅ ↔ ∀𝑥𝐵 ¬ 𝑥𝐴)
42, 3bitri 263 1 ((𝐴𝐵) = ∅ ↔ ∀𝑥𝐵 ¬ 𝑥𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 195   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∩ cin 3539  ∅c0 3874 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-v 3175  df-dif 3543  df-in 3547  df-nul 3875 This theorem is referenced by:  zfreg2OLD  8386  kqdisj  21345  iccntr  22432  ntrneicls11  37408  iooinlbub  38570  stoweidlem57  38950
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