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Theorem disjr 4364
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 4115 . . 3 (𝐴𝐵) = (𝐵𝐴)
21eqeq1i 2742 . 2 ((𝐴𝐵) = ∅ ↔ (𝐵𝐴) = ∅)
3 disj 4362 . 2 ((𝐵𝐴) = ∅ ↔ ∀𝑥𝐵 ¬ 𝑥𝐴)
42, 3bitri 278 1 ((𝐴𝐵) = ∅ ↔ ∀𝑥𝐵 ¬ 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209   = wceq 1543  wcel 2110  wral 3061  cin 3865  c0 4237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rab 3070  df-dif 3869  df-in 3873  df-nul 4238
This theorem is referenced by:  kqdisj  22629  iccntr  23718  numedglnl  27235  fmlasucdisj  33074  ntrneicls11  41377  iooinlbub  42714  stoweidlem57  43273
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