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Theorem disj 3995
Description: Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 17-Feb-2004.)
Assertion
Ref Expression
disj ((𝐴𝐵) = ∅ ↔ ∀𝑥𝐴 ¬ 𝑥𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem disj
StepHypRef Expression
1 df-in 3567 . . . 4 (𝐴𝐵) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
21eqeq1i 2626 . . 3 ((𝐴𝐵) = ∅ ↔ {𝑥 ∣ (𝑥𝐴𝑥𝐵)} = ∅)
3 abeq1 2730 . . 3 ({𝑥 ∣ (𝑥𝐴𝑥𝐵)} = ∅ ↔ ∀𝑥((𝑥𝐴𝑥𝐵) ↔ 𝑥 ∈ ∅))
4 imnan 438 . . . . 5 ((𝑥𝐴 → ¬ 𝑥𝐵) ↔ ¬ (𝑥𝐴𝑥𝐵))
5 noel 3901 . . . . . 6 ¬ 𝑥 ∈ ∅
65nbn 362 . . . . 5 (¬ (𝑥𝐴𝑥𝐵) ↔ ((𝑥𝐴𝑥𝐵) ↔ 𝑥 ∈ ∅))
74, 6bitr2i 265 . . . 4 (((𝑥𝐴𝑥𝐵) ↔ 𝑥 ∈ ∅) ↔ (𝑥𝐴 → ¬ 𝑥𝐵))
87albii 1744 . . 3 (∀𝑥((𝑥𝐴𝑥𝐵) ↔ 𝑥 ∈ ∅) ↔ ∀𝑥(𝑥𝐴 → ¬ 𝑥𝐵))
92, 3, 83bitri 286 . 2 ((𝐴𝐵) = ∅ ↔ ∀𝑥(𝑥𝐴 → ¬ 𝑥𝐵))
10 df-ral 2913 . 2 (∀𝑥𝐴 ¬ 𝑥𝐵 ↔ ∀𝑥(𝑥𝐴 → ¬ 𝑥𝐵))
119, 10bitr4i 267 1 ((𝐴𝐵) = ∅ ↔ ∀𝑥𝐴 ¬ 𝑥𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  wal 1478   = wceq 1480  wcel 1987  {cab 2607  wral 2908  cin 3559  c0 3897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-v 3192  df-dif 3563  df-in 3567  df-nul 3898
This theorem is referenced by:  disjr  3996  disj1  3997  disjne  4000  otiunsndisj  4950  onxpdisj  5816  f0rn0  6057  onint  6957  zfreg  8460  zfregOLD  8462  kmlem4  8935  fin23lem30  9124  fin23lem31  9125  isf32lem3  9137  fpwwe2  9425  renfdisj  10058  fvinim0ffz  12543  s3iunsndisj  13657  metdsge  22592  2wspdisj  26757  2wspiundisj  26758  2wspmdisj  27093  subfacp1lem1  30922  dfpo2  31406  dvmptfprodlem  39496  stoweidlem26  39580  stoweidlem59  39613  iundjiunlem  40013  otiunsndisjX  40625
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