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Theorem disj1 4450
Description: Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 19-Aug-1993.)
Assertion
Ref Expression
disj1 ((𝐴𝐵) = ∅ ↔ ∀𝑥(𝑥𝐴 → ¬ 𝑥𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem disj1
StepHypRef Expression
1 disj 4447 . 2 ((𝐴𝐵) = ∅ ↔ ∀𝑥𝐴 ¬ 𝑥𝐵)
2 df-ral 3061 . 2 (∀𝑥𝐴 ¬ 𝑥𝐵 ↔ ∀𝑥(𝑥𝐴 → ¬ 𝑥𝐵))
31, 2bitri 275 1 ((𝐴𝐵) = ∅ ↔ ∀𝑥(𝑥𝐴 → ¬ 𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wal 1538   = wceq 1540  wcel 2105  wral 3060  cin 3947  c0 4322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-dif 3951  df-in 3955  df-nul 4323
This theorem is referenced by:  reldisj  4451  reldisjOLD  4452  disj3  4453  undif4  4466  disjsn  4715  funun  6594  zfregs2  9734  dfac5lem4  10127  isf32lem9  10362  fzodisj  13673  fzodisjsn  13677  inpr0  32051  bnj1280  34344  ecin0  37537  zfregs2VD  43917
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