Theorem disjsn 1836

Description: Intersection with singleton of non-member is disjoint.

Assertion

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

Proof of Theorem disjsn

Step	Hyp	Ref	Expression
1		noel 1711	. . . 4 ⊢ ¬ B ∈ ∅
2		eleq2 1150	. . . 4 ⊢ ((A ∩ {B}) = ∅ → (B ∈ (A ∩ {B}) ↔ B ∈ ∅))
3	1, 2	mtbiri 539	. . 3 ⊢ ((A ∩ {B}) = ∅ → ¬ B ∈ (A ∩ {B}))
4		snidg 1828	. . . . 5 ⊢ (B ∈ A → B ∈ {B})
5	4	ancli 244	. . . 4 ⊢ (B ∈ A → (B ∈ A ∧ B ∈ {B}))
6		elin 1635	. . . 4 ⊢ (B ∈ (A ∩ {B}) ↔ (B ∈ A ∧ B ∈ {B}))
7	5, 6	sylibr 175	. . 3 ⊢ (B ∈ A → B ∈ (A ∩ {B}))
8	3, 7	nsyl 102	. 2 ⊢ ((A ∩ {B}) = ∅ → ¬ B ∈ A)
9		eleq1 1149	. . . . . . . 8 ⊢ (x = B → (x ∈ A ↔ B ∈ A))
10	9	biimpcd 137	. . . . . . 7 ⊢ (x ∈ A → (x = B → B ∈ A))
11		elsn 1820	. . . . . . 7 ⊢ (x ∈ {B} ↔ x = B)
12	10, 11	syl5ib 181	. . . . . 6 ⊢ (x ∈ A → (x ∈ {B} → B ∈ A))
13	12	con3d 87	. . . . 5 ⊢ (x ∈ A → (¬ B ∈ A → ¬ x ∈ {B}))
14	13	com12 13	. . . 4 ⊢ (¬ B ∈ A → (x ∈ A → ¬ x ∈ {B}))
15	14	19.21aiv 943	. . 3 ⊢ (¬ B ∈ A → ∀x(x ∈ A → ¬ x ∈ {B}))
16		disj1 1734	. . 3 ⊢ ((A ∩ {B}) = ∅ ↔ ∀x(x ∈ A → ¬ x ∈ {B}))
17	15, 16	sylibr 175	. 2 ⊢ (¬ B ∈ A → (A ∩ {B}) = ∅)
18	8, 17	impbi 139	1 ⊢ ((A ∩ {B}) = ∅ ↔ ¬ B ∈ A)

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672   = wceq 1091   ∈ wcel 1092   ∩ cin 1486  ∅c0 1707  {csn 1808

This theorem is referenced by:  disjsn2 1837  orddisj 2236  ndmima 2623  limensuci 3401  php 3409  infensuc 3484  kmlem2 3581  facnnt 4870  fac0 4871

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-nul 1708  df-sn 1811  df-pr 1812

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