Theorem disj3 4387

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		pm4.71 558	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)))
2		eldif 3897	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
3	2	bibi2i 338	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ (𝐴 ∖ 𝐵)) ↔ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)))
4	1, 3	bitr4i 277	. . 3 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ (𝐴 ∖ 𝐵)))
5	4	albii 1822	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ (𝐴 ∖ 𝐵)))
6		disj1 4384	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵))
7		dfcleq 2731	. 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 205   ∧ wa 396  ∀wal 1537   = wceq 1539   ∈ wcel 2106   ∖ cdif 3884   ∩ cin 3886  ∅c0 4256

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-v 3434  df-dif 3890  df-in 3894  df-nul 4257

This theorem is referenced by:  disjel  4390  disj4  4392  uneqdifeq  4423  difprsn1  4733  diftpsn3  4735  ssunsn2  4760  orddif  6359  php  8993  phpOLD  9005  hartogslem1  9301  infeq5i  9394  cantnfp1lem3  9438  dju1dif  9928  infdju1  9945  ssxr  11044  dprd2da  19645  dmdprdsplit2lem  19648  ablfac1eulem  19675  lbsextlem4  20423  opsrtoslem2  21263  alexsublem  23195  volun  24709  lhop1lem  25177  ex-dif  28787  difeq  30865  imadifxp  30940  disjdsct  31035  fzodif1  31114  carsgclctunlem1  32284  probun  32386  ballotlemfp1  32458  bj-disj2r  35218  topdifinfeq  35521  finixpnum  35762  lindsadd  35770  poimirlem11  35788  poimirlem12  35789  poimirlem13  35790  poimirlem14  35791  poimirlem16  35793  poimirlem18  35795  poimirlem21  35798  poimirlem22  35799  poimirlem27  35804  asindmre  35860  kelac2  40890  pwfi2f1o  40921  iccdifioo  43053  iccdifprioo  43054