Theorem difin0ss 1753

Description: Difference, intersection, and subclass relationship.

Assertion

Ref	Expression
difin0ss	⊢ (((A ∖ B) ∩ C) = ∅ → (C ⊆ A → C ⊆ B))

Proof of Theorem difin0ss

Step	Hyp	Ref	Expression
1		eq0 1719	. . 3 ⊢ (((A ∖ B) ∩ C) = ∅ ↔ ∀x ¬ x ∈ ((A ∖ B) ∩ C))
2		annim 206	. . . . . . . . 9 ⊢ ((x ∈ A ∧ ¬ x ∈ B) ↔ ¬ (x ∈ A → x ∈ B))
3	2	anbi2i 367	. . . . . . . 8 ⊢ ((x ∈ C ∧ (x ∈ A ∧ ¬ x ∈ B)) ↔ (x ∈ C ∧ ¬ (x ∈ A → x ∈ B)))
4		ancom 333	. . . . . . . 8 ⊢ ((x ∈ C ∧ (x ∈ A ∧ ¬ x ∈ B)) ↔ ((x ∈ A ∧ ¬ x ∈ B) ∧ x ∈ C))
5	3, 4	bitr3 153	. . . . . . 7 ⊢ ((x ∈ C ∧ ¬ (x ∈ A → x ∈ B)) ↔ ((x ∈ A ∧ ¬ x ∈ B) ∧ x ∈ C))
6	5	negbii 162	. . . . . 6 ⊢ (¬ (x ∈ C ∧ ¬ (x ∈ A → x ∈ B)) ↔ ¬ ((x ∈ A ∧ ¬ x ∈ B) ∧ x ∈ C))
7		iman 205	. . . . . 6 ⊢ ((x ∈ C → (x ∈ A → x ∈ B)) ↔ ¬ (x ∈ C ∧ ¬ (x ∈ A → x ∈ B)))
8		elin 1635	. . . . . . . 8 ⊢ (x ∈ ((A ∖ B) ∩ C) ↔ (x ∈ (A ∖ B) ∧ x ∈ C))
9		eldif 1496	. . . . . . . . 9 ⊢ (x ∈ (A ∖ B) ↔ (x ∈ A ∧ ¬ x ∈ B))
10	9	anbi1i 368	. . . . . . . 8 ⊢ ((x ∈ (A ∖ B) ∧ x ∈ C) ↔ ((x ∈ A ∧ ¬ x ∈ B) ∧ x ∈ C))
11	8, 10	bitr 151	. . . . . . 7 ⊢ (x ∈ ((A ∖ B) ∩ C) ↔ ((x ∈ A ∧ ¬ x ∈ B) ∧ x ∈ C))
12	11	negbii 162	. . . . . 6 ⊢ (¬ x ∈ ((A ∖ B) ∩ C) ↔ ¬ ((x ∈ A ∧ ¬ x ∈ B) ∧ x ∈ C))
13	6, 7, 12	3bitr4 158	. . . . 5 ⊢ ((x ∈ C → (x ∈ A → x ∈ B)) ↔ ¬ x ∈ ((A ∖ B) ∩ C))
14		ax-2 4	. . . . 5 ⊢ ((x ∈ C → (x ∈ A → x ∈ B)) → ((x ∈ C → x ∈ A) → (x ∈ C → x ∈ B)))
15	13, 14	sylbir 176	. . . 4 ⊢ (¬ x ∈ ((A ∖ B) ∩ C) → ((x ∈ C → x ∈ A) → (x ∈ C → x ∈ B)))
16	15	19.20ii 692	. . 3 ⊢ (∀x ¬ x ∈ ((A ∖ B) ∩ C) → (∀x(x ∈ C → x ∈ A) → ∀x(x ∈ C → x ∈ B)))
17	1, 16	sylbi 174	. 2 ⊢ (((A ∖ B) ∩ C) = ∅ → (∀x(x ∈ C → x ∈ A) → ∀x(x ∈ C → x ∈ B)))
18		dfss2 1497	. 2 ⊢ (C ⊆ A ↔ ∀x(x ∈ C → x ∈ A))
19		dfss2 1497	. 2 ⊢ (C ⊆ B ↔ ∀x(x ∈ C → x ∈ B))
20	17, 18, 19	3imtr4g 426	1 ⊢ (((A ∖ B) ∩ C) = ∅ → (C ⊆ A → C ⊆ B))

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

This theorem is referenced by:  tz7.7 2224  tfi 2244

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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