Theorem disj3 4408

Description: Two ways of saying that two classes are disjoint. (Contributed by NM, 19-May-1998.)

Assertion

Ref	Expression
disj3	⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 = (𝐴 ∖ 𝐵))

Proof of Theorem disj3

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pm4.71 557	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)))
2		eldif 3913	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
3	2	bibi2i 337	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ (𝐴 ∖ 𝐵)) ↔ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)))
4	1, 3	bitr4i 278	. . 3 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ (𝐴 ∖ 𝐵)))
5	4	albii 1821	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ (𝐴 ∖ 𝐵)))
6		disj1 4406	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵))
7		dfcleq 2730	. 2 ⊢ (𝐴 = (𝐴 ∖ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ (𝐴 ∖ 𝐵)))
8	5, 6, 7	3bitr4i 303	1 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 = (𝐴 ∖ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540   = wceq 1542   ∈ wcel 2114   ∖ cdif 3900   ∩ cin 3902  ∅c0 4287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-v 3444  df-dif 3906  df-in 3910  df-nul 4288

This theorem is referenced by:  disjel  4411  disj4  4413  uneqdifeq  4447  difprsn1  4758  diftpsn3  4760  ssunsn2  4785  orddif  6423  php  9143  hartogslem1  9459  infeq5i  9557  cantnfp1lem3  9601  dju1dif  10095  infdju1  10112  ssxr  11214  dprd2da  19985  dmdprdsplit2lem  19988  ablfac1eulem  20015  lbsextlem4  21128  opsrtoslem2  22023  alexsublem  24000  volun  25514  lhop1lem  25986  ex-dif  30510  difeq  32605  imadifxp  32688  disjdsct  32793  fzodif1  32883  carsgclctunlem1  34495  probun  34597  ballotlemfp1  34670  bj-disj2r  37276  topdifinfeq  37605  finixpnum  37856  lindsadd  37864  poimirlem11  37882  poimirlem12  37883  poimirlem13  37884  poimirlem14  37885  poimirlem16  37887  poimirlem18  37889  poimirlem21  37892  poimirlem22  37893  poimirlem27  37898  asindmre  37954  kelac2  43422  pwfi2f1o  43453  iccdifioo  45875  iccdifprioo  45876