Theorem disj 3592

Description: Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 17-Feb-2004.)

Assertion

Ref	Expression
disj	⊢ ((A ∩ B) = ∅ ↔ ∀x ∈ A ¬ x ∈ B)

Distinct variable groups:   x,A   x,B

Proof of Theorem disj

Step	Hyp	Ref	Expression
1		elin 3220	. . . . 5 ⊢ (x ∈ (A ∩ B) ↔ (x ∈ A ∧ x ∈ B))
2		df-an 360	. . . . 5 ⊢ ((x ∈ A ∧ x ∈ B) ↔ ¬ (x ∈ A → ¬ x ∈ B))
3	1, 2	bitr2i 241	. . . 4 ⊢ (¬ (x ∈ A → ¬ x ∈ B) ↔ x ∈ (A ∩ B))
4	3	con1bii 321	. . 3 ⊢ (¬ x ∈ (A ∩ B) ↔ (x ∈ A → ¬ x ∈ B))
5	4	albii 1566	. 2 ⊢ (∀x ¬ x ∈ (A ∩ B) ↔ ∀x(x ∈ A → ¬ x ∈ B))
6		eq0 3565	. 2 ⊢ ((A ∩ B) = ∅ ↔ ∀x ¬ x ∈ (A ∩ B))
7		df-ral 2620	. 2 ⊢ (∀x ∈ A ¬ x ∈ B ↔ ∀x(x ∈ A → ¬ x ∈ B))
8	5, 6, 7	3bitr4i 268	1 ⊢ ((A ∩ B) = ∅ ↔ ∀x ∈ A ¬ x ∈ B)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  ∀wral 2615   ∩ 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:  disjr  3593  disj1  3594  disjne  3597  disj5  3891  pw10  4162  pw1disj  4168  phidisjnn  4616  fvun1  5380  xpnedisj  5514  disjex  5824