Theorem difin0ss 2342

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 2304	. . 3 ⊢ (((A ∖ B) ∩ C) = ∅ ↔ ∀x ¬ x ∈ ((A ∖ B) ∩ C))
2		annim 238	. . . . . . . . 9 ⊢ ((x ∈ A ⋀ ¬ x ∈ B) ↔ ¬ (x ∈ A → x ∈ B))
3	2	anbi2i 483	. . . . . . . 8 ⊢ ((x ∈ C ⋀ (x ∈ A ⋀ ¬ x ∈ B)) ↔ (x ∈ C ⋀ ¬ (x ∈ A → x ∈ B)))
4		ancom 438	. . . . . . . 8 ⊢ ((x ∈ C ⋀ (x ∈ A ⋀ ¬ x ∈ B)) ↔ ((x ∈ A ⋀ ¬ x ∈ B) ⋀ x ∈ C))
5	3, 4	bitr3 175	. . . . . . 7 ⊢ ((x ∈ C ⋀ ¬ (x ∈ A → x ∈ B)) ↔ ((x ∈ A ⋀ ¬ x ∈ B) ⋀ x ∈ C))
6	5	negbii 187	. . . . . 6 ⊢ (¬ (x ∈ C ⋀ ¬ (x ∈ A → x ∈ B)) ↔ ¬ ((x ∈ A ⋀ ¬ x ∈ B) ⋀ x ∈ C))
7		iman 237	. . . . . 6 ⊢ ((x ∈ C → (x ∈ A → x ∈ B)) ↔ ¬ (x ∈ C ⋀ ¬ (x ∈ A → x ∈ B)))
8		elin 2216	. . . . . . . 8 ⊢ (x ∈ ((A ∖ B) ∩ C) ↔ (x ∈ (A ∖ B) ⋀ x ∈ C))
9		eldif 2066	. . . . . . . . 9 ⊢ (x ∈ (A ∖ B) ↔ (x ∈ A ⋀ ¬ x ∈ B))
10	9	anbi1i 484	. . . . . . . 8 ⊢ ((x ∈ (A ∖ B) ⋀ x ∈ C) ↔ ((x ∈ A ⋀ ¬ x ∈ B) ⋀ x ∈ C))
11	8, 10	bitr 173	. . . . . . 7 ⊢ (x ∈ ((A ∖ B) ∩ C) ↔ ((x ∈ A ⋀ ¬ x ∈ B) ⋀ x ∈ C))
12	11	negbii 187	. . . . . 6 ⊢ (¬ x ∈ ((A ∖ B) ∩ C) ↔ ¬ ((x ∈ A ⋀ ¬ x ∈ B) ⋀ x ∈ C))
13	6, 7, 12	3bitr4 183	. . . . 5 ⊢ ((x ∈ C → (x ∈ A → x ∈ B)) ↔ ¬ x ∈ ((A ∖ B) ∩ C))
14		ax-2 5	. . . . 5 ⊢ ((x ∈ C → (x ∈ A → x ∈ B)) → ((x ∈ C → x ∈ A) → (x ∈ C → x ∈ B)))
15	13, 14	sylbir 201	. . . 4 ⊢ (¬ x ∈ ((A ∖ B) ∩ C) → ((x ∈ C → x ∈ A) → (x ∈ C → x ∈ B)))
16	15	19.20ii 1001	. . 3 ⊢ (∀x ¬ x ∈ ((A ∖ B) ∩ C) → (∀x(x ∈ C → x ∈ A) → ∀x(x ∈ C → x ∈ B)))
17	1, 16	sylbi 199	. 2 ⊢ (((A ∖ B) ∩ C) = ∅ → (∀x(x ∈ C → x ∈ A) → ∀x(x ∈ C → x ∈ B)))
18		dfss2 2067	. 2 ⊢ (C ⊆ A ↔ ∀x(x ∈ C → x ∈ A))
19		dfss2 2067	. 2 ⊢ (C ⊆ B ↔ ∀x(x ∈ C → x ∈ B))
20	17, 18, 19	3imtr4g 556	1 ⊢ (((A ∖ B) ∩ C) = ∅ → (C ⊆ A → C ⊆ B))

Syntax hints:  ¬ wn 2   → wi 3   ⋀ wa 223  ∀wal 958   = wceq 960   ∈ wcel 962   ∖ cdif 2053   ∩ cin 2055   ⊆ wss 2056  ∅c0 2289

This theorem is referenced by:  tz7.7 2987  tfi 3140

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-10 970  ax-12 972  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1129  ax-10o 1146  ax-16 1216  ax-11o 1224  ax-ext 1466

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 985  df-sb 1178  df-clab 1471  df-cleq 1476  df-clel 1479  df-ne 1594  df-v 1819  df-dif 2058  df-in 2060  df-ss 2062  df-nul 2290

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