Theorem difin0ss 3900

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 3888	. 2 ⊢ (((𝐴 ∖ 𝐵) ∩ 𝐶) = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ ((𝐴 ∖ 𝐵) ∩ 𝐶))
2		iman 439	. . . . . 6 ⊢ ((𝑥 ∈ 𝐶 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) ↔ ¬ (𝑥 ∈ 𝐶 ∧ ¬ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)))
3		elin 3758	. . . . . . 7 ⊢ (𝑥 ∈ ((𝐴 ∖ 𝐵) ∩ 𝐶) ↔ (𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥 ∈ 𝐶))
4		eldif 3550	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
5	4	anbi2ci 728	. . . . . . 7 ⊢ ((𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)))
6		annim 440	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ↔ ¬ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
7	6	anbi2i 726	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) ↔ (𝑥 ∈ 𝐶 ∧ ¬ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)))
8	3, 5, 7	3bitri 285	. . . . . 6 ⊢ (𝑥 ∈ ((𝐴 ∖ 𝐵) ∩ 𝐶) ↔ (𝑥 ∈ 𝐶 ∧ ¬ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)))
9	2, 8	xchbinxr 324	. . . . 5 ⊢ ((𝑥 ∈ 𝐶 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) ↔ ¬ 𝑥 ∈ ((𝐴 ∖ 𝐵) ∩ 𝐶))
10		ax-2 7	. . . . 5 ⊢ ((𝑥 ∈ 𝐶 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) → ((𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐵)))
11	9, 10	sylbir 224	. . . 4 ⊢ (¬ 𝑥 ∈ ((𝐴 ∖ 𝐵) ∩ 𝐶) → ((𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐵)))
12	11	al2imi 1733	. . 3 ⊢ (∀𝑥 ¬ 𝑥 ∈ ((𝐴 ∖ 𝐵) ∩ 𝐶) → (∀𝑥(𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐴) → ∀𝑥(𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐵)))
13		dfss2 3557	. . 3 ⊢ (𝐶 ⊆ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐴))
14		dfss2 3557	. . 3 ⊢ (𝐶 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐵))
15	12, 13, 14	3imtr4g 284	. 2 ⊢ (∀𝑥 ¬ 𝑥 ∈ ((𝐴 ∖ 𝐵) ∩ 𝐶) → (𝐶 ⊆ 𝐴 → 𝐶 ⊆ 𝐵))
16	1, 15	sylbi 206	1 ⊢ (((𝐴 ∖ 𝐵) ∩ 𝐶) = ∅ → (𝐶 ⊆ 𝐴 → 𝐶 ⊆ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875

This theorem is referenced by:  tz7.7  5666  tfi  6945  lebnumlem3  22570