Theorem inssdif0 1754

Description: Intersection, subclass, and difference relationship.

Assertion

Ref	Expression
inssdif0	⊢ ((A ∩ B) ⊆ C ↔ (A ∩ (B ∖ C)) = ∅)

Proof of Theorem inssdif0

Step	Hyp	Ref	Expression
1		impexp 276	. . . . 5 ⊢ (((x ∈ A ∧ x ∈ B) → x ∈ C) ↔ (x ∈ A → (x ∈ B → x ∈ C)))
2		iman 205	. . . . . 6 ⊢ ((x ∈ B → x ∈ C) ↔ ¬ (x ∈ B ∧ ¬ x ∈ C))
3	2	imbi2i 160	. . . . 5 ⊢ ((x ∈ A → (x ∈ B → x ∈ C)) ↔ (x ∈ A → ¬ (x ∈ B ∧ ¬ x ∈ C)))
4		imnan 207	. . . . 5 ⊢ ((x ∈ A → ¬ (x ∈ B ∧ ¬ x ∈ C)) ↔ ¬ (x ∈ A ∧ (x ∈ B ∧ ¬ x ∈ C)))
5	1, 3, 4	3bitr 155	. . . 4 ⊢ (((x ∈ A ∧ x ∈ B) → x ∈ C) ↔ ¬ (x ∈ A ∧ (x ∈ B ∧ ¬ x ∈ C)))
6		elin 1635	. . . . 5 ⊢ (x ∈ (A ∩ B) ↔ (x ∈ A ∧ x ∈ B))
7	6	imbi1i 161	. . . 4 ⊢ ((x ∈ (A ∩ B) → x ∈ C) ↔ ((x ∈ A ∧ x ∈ B) → x ∈ C))
8		elin 1635	. . . . . 6 ⊢ (x ∈ (A ∩ (B ∖ C)) ↔ (x ∈ A ∧ x ∈ (B ∖ C)))
9		eldif 1496	. . . . . . 7 ⊢ (x ∈ (B ∖ C) ↔ (x ∈ B ∧ ¬ x ∈ C))
10	9	anbi2i 367	. . . . . 6 ⊢ ((x ∈ A ∧ x ∈ (B ∖ C)) ↔ (x ∈ A ∧ (x ∈ B ∧ ¬ x ∈ C)))
11	8, 10	bitr 151	. . . . 5 ⊢ (x ∈ (A ∩ (B ∖ C)) ↔ (x ∈ A ∧ (x ∈ B ∧ ¬ x ∈ C)))
12	11	negbii 162	. . . 4 ⊢ (¬ x ∈ (A ∩ (B ∖ C)) ↔ ¬ (x ∈ A ∧ (x ∈ B ∧ ¬ x ∈ C)))
13	5, 7, 12	3bitr4 158	. . 3 ⊢ ((x ∈ (A ∩ B) → x ∈ C) ↔ ¬ x ∈ (A ∩ (B ∖ C)))
14	13	bial 695	. 2 ⊢ (∀x(x ∈ (A ∩ B) → x ∈ C) ↔ ∀x ¬ x ∈ (A ∩ (B ∖ C)))
15		dfss2 1497	. 2 ⊢ ((A ∩ B) ⊆ C ↔ ∀x(x ∈ (A ∩ B) → x ∈ C))
16		eq0 1719	. 2 ⊢ ((A ∩ (B ∖ C)) = ∅ ↔ ∀x ¬ x ∈ (A ∩ (B ∖ C)))
17	14, 15, 16	3bitr4 158	1 ⊢ ((A ∩ B) ⊆ C ↔ (A ∩ (B ∖ C)) = ∅)

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672   = wceq 1091   ∈ wcel 1092   ∖ cdif 1484   ∩ cin 1486   ⊆ wss 1487  ∅c0 1707

This theorem is referenced by:  difdisj 1758  inf3lem3 3466

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-v 1349  df-dif 1489  df-in 1491  df-ss 1492  df-nul 1708

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