MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  disj3 Structured version   Visualization version   GIF version

Theorem disj3 4403
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 pm4.71 560 . . . 4 ((𝑥𝐴 → ¬ 𝑥𝐵) ↔ (𝑥𝐴 ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵)))
2 eldif 3946 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
32bibi2i 340 . . . 4 ((𝑥𝐴𝑥 ∈ (𝐴𝐵)) ↔ (𝑥𝐴 ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵)))
41, 3bitr4i 280 . . 3 ((𝑥𝐴 → ¬ 𝑥𝐵) ↔ (𝑥𝐴𝑥 ∈ (𝐴𝐵)))
54albii 1816 . 2 (∀𝑥(𝑥𝐴 → ¬ 𝑥𝐵) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ (𝐴𝐵)))
6 disj1 4401 . 2 ((𝐴𝐵) = ∅ ↔ ∀𝑥(𝑥𝐴 → ¬ 𝑥𝐵))
7 dfcleq 2815 . 2 (𝐴 = (𝐴𝐵) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ (𝐴𝐵)))
85, 6, 73bitr4i 305 1 ((𝐴𝐵) = ∅ ↔ 𝐴 = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  wal 1531   = wceq 1533  wcel 2110  cdif 3933  cin 3935  c0 4291
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-v 3497  df-dif 3939  df-in 3943  df-nul 4292
This theorem is referenced by:  disjel  4406  disj4  4408  uneqdifeq  4438  difprsn1  4727  diftpsn3  4729  ssunsn2  4754  orddif  6279  php  8695  hartogslem1  9000  infeq5i  9093  cantnfp1lem3  9137  dju1dif  9592  infdju1  9609  ssxr  10704  dprd2da  19158  dmdprdsplit2lem  19161  ablfac1eulem  19188  lbsextlem4  19927  opsrtoslem2  20259  alexsublem  22646  volun  24140  lhop1lem  24604  ex-dif  28196  difeq  30274  imadifxp  30345  disjdsct  30432  fzodif1  30510  carsgclctunlem1  31570  probun  31672  ballotlemfp1  31744  bj-disj2r  34335  topdifinfeq  34625  finixpnum  34871  lindsadd  34879  poimirlem11  34897  poimirlem12  34898  poimirlem13  34899  poimirlem14  34900  poimirlem16  34902  poimirlem18  34904  poimirlem21  34907  poimirlem22  34908  poimirlem27  34913  asindmre  34971  dffltz  39264  kelac2  39658  pwfi2f1o  39689  iccdifioo  41783  iccdifprioo  41784
  Copyright terms: Public domain W3C validator