Theorem disj3 3596

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		pm4.71 611	. . . 4 ⊢ ((x ∈ A → ¬ x ∈ B) ↔ (x ∈ A ↔ (x ∈ A ∧ ¬ x ∈ B)))
2		eldif 3222	. . . . 5 ⊢ (x ∈ (A ∖ B) ↔ (x ∈ A ∧ ¬ x ∈ B))
3	2	bibi2i 304	. . . 4 ⊢ ((x ∈ A ↔ x ∈ (A ∖ B)) ↔ (x ∈ A ↔ (x ∈ A ∧ ¬ x ∈ B)))
4	1, 3	bitr4i 243	. . 3 ⊢ ((x ∈ A → ¬ x ∈ B) ↔ (x ∈ A ↔ x ∈ (A ∖ B)))
5	4	albii 1566	. 2 ⊢ (∀x(x ∈ A → ¬ x ∈ B) ↔ ∀x(x ∈ A ↔ x ∈ (A ∖ B)))
6		disj1 3594	. 2 ⊢ ((A ∩ B) = ∅ ↔ ∀x(x ∈ A → ¬ x ∈ B))
7		dfcleq 2347	. 2 ⊢ (A = (A ∖ B) ↔ ∀x(x ∈ A ↔ x ∈ (A ∖ B)))
8	5, 6, 7	3bitr4i 268	1 ⊢ ((A ∩ B) = ∅ ↔ A = (A ∖ B))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710   ∖ cdif 3207   ∩ cin 3209  ∅c0 3551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552

This theorem is referenced by:  disjel  3598  disj4  3600  uneqdifeq  3639  difprsn1  3848  diftpsn3  3850  ssunsn2  3866