Theorem difin0ss 3616

Description: Difference, intersection, and subclass relationship. (Contributed by NM, 30-Apr-1994.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)

Assertion

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

Proof of Theorem difin0ss

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eq0 3564	. 2 ⊢ (((A ∖ B) ∩ C) = ∅ ↔ ∀x ¬ x ∈ ((A ∖ B) ∩ C))
2		iman 413	. . . . . 6 ⊢ ((x ∈ C → (x ∈ A → x ∈ B)) ↔ ¬ (x ∈ C ∧ ¬ (x ∈ A → x ∈ B)))
3		elin 3219	. . . . . . . 8 ⊢ (x ∈ ((A ∖ B) ∩ C) ↔ (x ∈ (A ∖ B) ∧ x ∈ C))
4		eldif 3221	. . . . . . . . 9 ⊢ (x ∈ (A ∖ B) ↔ (x ∈ A ∧ ¬ x ∈ B))
5	4	anbi1i 676	. . . . . . . 8 ⊢ ((x ∈ (A ∖ B) ∧ x ∈ C) ↔ ((x ∈ A ∧ ¬ x ∈ B) ∧ x ∈ C))
6	3, 5	bitri 240	. . . . . . 7 ⊢ (x ∈ ((A ∖ B) ∩ C) ↔ ((x ∈ A ∧ ¬ x ∈ B) ∧ x ∈ C))
7		ancom 437	. . . . . . 7 ⊢ ((x ∈ C ∧ (x ∈ A ∧ ¬ x ∈ B)) ↔ ((x ∈ A ∧ ¬ x ∈ B) ∧ x ∈ C))
8		annim 414	. . . . . . . 8 ⊢ ((x ∈ A ∧ ¬ x ∈ B) ↔ ¬ (x ∈ A → x ∈ B))
9	8	anbi2i 675	. . . . . . 7 ⊢ ((x ∈ C ∧ (x ∈ A ∧ ¬ x ∈ B)) ↔ (x ∈ C ∧ ¬ (x ∈ A → x ∈ B)))
10	6, 7, 9	3bitr2i 264	. . . . . 6 ⊢ (x ∈ ((A ∖ B) ∩ C) ↔ (x ∈ C ∧ ¬ (x ∈ A → x ∈ B)))
11	2, 10	xchbinxr 302	. . . . 5 ⊢ ((x ∈ C → (x ∈ A → x ∈ B)) ↔ ¬ x ∈ ((A ∖ B) ∩ C))
12		ax-2 7	. . . . 5 ⊢ ((x ∈ C → (x ∈ A → x ∈ B)) → ((x ∈ C → x ∈ A) → (x ∈ C → x ∈ B)))
13	11, 12	sylbir 204	. . . 4 ⊢ (¬ x ∈ ((A ∖ B) ∩ C) → ((x ∈ C → x ∈ A) → (x ∈ C → x ∈ B)))
14	13	al2imi 1561	. . 3 ⊢ (∀x ¬ x ∈ ((A ∖ B) ∩ C) → (∀x(x ∈ C → x ∈ A) → ∀x(x ∈ C → x ∈ B)))
15		dfss2 3262	. . 3 ⊢ (C ⊆ A ↔ ∀x(x ∈ C → x ∈ A))
16		dfss2 3262	. . 3 ⊢ (C ⊆ B ↔ ∀x(x ∈ C → x ∈ B))
17	14, 15, 16	3imtr4g 261	. 2 ⊢ (∀x ¬ x ∈ ((A ∖ B) ∩ C) → (C ⊆ A → C ⊆ B))
18	1, 17	sylbi 187	1 ⊢ (((A ∖ B) ∩ C) = ∅ → (C ⊆ A → C ⊆ B))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710   ∖ cdif 3206   ∩ cin 3208   ⊆ wss 3257  ∅c0 3550

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 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259  df-nul 3551

This theorem is referenced by: (None)