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Theorem disj3 3999
Description: Two ways of saying that two classes are disjoint. (Contributed by NM, 19-May-1998.)
Assertion
Ref Expression
disj3 ((𝐴𝐵) = ∅ ↔ 𝐴 = (𝐴𝐵))

Proof of Theorem disj3
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 pm4.71 661 . . . 4 ((𝑥𝐴 → ¬ 𝑥𝐵) ↔ (𝑥𝐴 ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵)))
2 eldif 3570 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
32bibi2i 327 . . . 4 ((𝑥𝐴𝑥 ∈ (𝐴𝐵)) ↔ (𝑥𝐴 ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵)))
41, 3bitr4i 267 . . 3 ((𝑥𝐴 → ¬ 𝑥𝐵) ↔ (𝑥𝐴𝑥 ∈ (𝐴𝐵)))
54albii 1744 . 2 (∀𝑥(𝑥𝐴 → ¬ 𝑥𝐵) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ (𝐴𝐵)))
6 disj1 3997 . 2 ((𝐴𝐵) = ∅ ↔ ∀𝑥(𝑥𝐴 → ¬ 𝑥𝐵))
7 dfcleq 2615 . 2 (𝐴 = (𝐴𝐵) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ (𝐴𝐵)))
85, 6, 73bitr4i 292 1 ((𝐴𝐵) = ∅ ↔ 𝐴 = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  wal 1478   = wceq 1480  wcel 1987  cdif 3557  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:  disjel  4001  disj4  4003  uneqdifeq  4035  uneqdifeqOLD  4036  difprsn1  4306  diftpsn3  4308  diftpsn3OLD  4309  ssunsn2  4334  orddif  5789  php  8104  hartogslem1  8407  infeq5i  8493  cantnfp1lem3  8537  cda1dif  8958  infcda1  8975  ssxr  10067  dprd2da  18381  dmdprdsplit2lem  18384  ablfac1eulem  18411  lbsextlem4  19101  opsrtoslem2  19425  alexsublem  21788  volun  23253  lhop1lem  23714  ex-dif  27168  difeq  29243  imadifxp  29300  disjdsct  29364  carsgclctunlem1  30202  probun  30304  ballotlemfp1  30376  bj-disj2r  32713  topdifinfeq  32869  finixpnum  33065  poimirlem11  33091  poimirlem12  33092  poimirlem13  33093  poimirlem14  33094  poimirlem16  33096  poimirlem18  33098  poimirlem21  33101  poimirlem22  33102  poimirlem27  33107  asindmre  33166  kelac2  37154  pwfi2f1o  37185  iccdifioo  39187  iccdifprioo  39188
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