Theorem disjsn 3787

Description: Intersection with the singleton of a non-member is disjoint. (Contributed by NM, 22-May-1998.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)

Assertion

Ref	Expression
disjsn	⊢ ((A ∩ {B}) = ∅ ↔ ¬ B ∈ A)

Proof of Theorem disjsn

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		disj1 3594	. 2 ⊢ ((A ∩ {B}) = ∅ ↔ ∀x(x ∈ A → ¬ x ∈ {B}))
2		con2b 324	. . . 4 ⊢ ((x ∈ A → ¬ x ∈ {B}) ↔ (x ∈ {B} → ¬ x ∈ A))
3		elsn 3749	. . . . 5 ⊢ (x ∈ {B} ↔ x = B)
4	3	imbi1i 315	. . . 4 ⊢ ((x ∈ {B} → ¬ x ∈ A) ↔ (x = B → ¬ x ∈ A))
5		imnan 411	. . . 4 ⊢ ((x = B → ¬ x ∈ A) ↔ ¬ (x = B ∧ x ∈ A))
6	2, 4, 5	3bitri 262	. . 3 ⊢ ((x ∈ A → ¬ x ∈ {B}) ↔ ¬ (x = B ∧ x ∈ A))
7	6	albii 1566	. 2 ⊢ (∀x(x ∈ A → ¬ x ∈ {B}) ↔ ∀x ¬ (x = B ∧ x ∈ A))
8		alnex 1543	. . 3 ⊢ (∀x ¬ (x = B ∧ x ∈ A) ↔ ¬ ∃x(x = B ∧ x ∈ A))
9		df-clel 2349	. . 3 ⊢ (B ∈ A ↔ ∃x(x = B ∧ x ∈ A))
10	8, 9	xchbinxr 302	. 2 ⊢ (∀x ¬ (x = B ∧ x ∈ A) ↔ ¬ B ∈ A)
11	1, 7, 10	3bitri 262	1 ⊢ ((A ∩ {B}) = ∅ ↔ ¬ B ∈ A)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ∩ cin 3209  ∅c0 3551  {csn 3738

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  df-sn 3742

This theorem is referenced by:  disjsn2  3788  ssunsn2  3866  dfiota4  4373  elsuc  4414  nndisjeq  4430  vfinncsp  4555  phiall  4619  ndmima  5026  fnunsn  5191  ressnop0  5437  1p1e2c  6156  2p1e3c  6157  nchoicelem7  6296  nchoicelem14  6303