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Theorem disj1 3264
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 3262 . 2 ((AB) = ∅ ↔ x A ¬ x B)
2 df-ral 2305 . 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 1240   = wceq 1242   wcel 1390  wral 2300  cin 2910  c0 3218
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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-dif 2914  df-in 2918  df-nul 3219
This theorem is referenced by:  reldisj  3265  disj3  3266  undif4  3278  disjsn  3423  funun  4887  fzodisj  8784
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