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Theorem disjr 4457
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 4217 . . 3 (𝐴𝐵) = (𝐵𝐴)
21eqeq1i 2740 . 2 ((𝐴𝐵) = ∅ ↔ (𝐵𝐴) = ∅)
3 disj 4456 . 2 ((𝐵𝐴) = ∅ ↔ ∀𝑥𝐵 ¬ 𝑥𝐴)
42, 3bitri 275 1 ((𝐴𝐵) = ∅ ↔ ∀𝑥𝐵 ¬ 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206   = wceq 1537  wcel 2106  wral 3059  cin 3962  c0 4339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rab 3434  df-dif 3966  df-in 3970  df-nul 4340
This theorem is referenced by:  kqdisj  23756  iccntr  24857  numedglnl  29176  fmlasucdisj  35384  ntrneicls11  44080  iooinlbub  45454  stoweidlem57  46013
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