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Theorem disj1 3247
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(x A → ¬ x B))
Distinct variable groups:   x,A   x,B

Proof of Theorem disj1
StepHypRef Expression
1 disj 3245 . 2 ((AB) = ∅ ↔ x A ¬ x B)
2 df-ral 2289 . 2 (x A ¬ x Bx(x A → ¬ x B))
31, 2bitri 173 1 ((AB) = ∅ ↔ x(x A → ¬ x B))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 98  wal 1226   = wceq 1228   wcel 1374  wral 2284  cin 2893  c0 3201
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-v 2537  df-dif 2897  df-in 2901  df-nul 3202
This theorem is referenced by:  reldisj  3248  disj3  3249  undif4  3261  disjsn  3406  funun  4870
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