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Theorem disjr 3382
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 3238 . . 3 (𝐴𝐵) = (𝐵𝐴)
21eqeq1i 2125 . 2 ((𝐴𝐵) = ∅ ↔ (𝐵𝐴) = ∅)
3 disj 3381 . 2 ((𝐵𝐴) = ∅ ↔ ∀𝑥𝐵 ¬ 𝑥𝐴)
42, 3bitri 183 1 ((𝐴𝐵) = ∅ ↔ ∀𝑥𝐵 ¬ 𝑥𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 104   = wceq 1316  wcel 1465  wral 2393  cin 3040  c0 3333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-v 2662  df-dif 3043  df-in 3047  df-nul 3334
This theorem is referenced by: (None)
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