Theorem difin0ss 4367

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 4342	. 2 ⊢ (((𝐴 ∖ 𝐵) ∩ 𝐶) = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ ((𝐴 ∖ 𝐵) ∩ 𝐶))
2		iman 402	. . . . . 6 ⊢ ((𝑥 ∈ 𝐶 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) ↔ ¬ (𝑥 ∈ 𝐶 ∧ ¬ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)))
3		elin 3963	. . . . . . 7 ⊢ (𝑥 ∈ ((𝐴 ∖ 𝐵) ∩ 𝐶) ↔ (𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥 ∈ 𝐶))
4		eldif 3957	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
5	4	anbi2ci 625	. . . . . . 7 ⊢ ((𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)))
6		annim 404	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ↔ ¬ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
7	6	anbi2i 623	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) ↔ (𝑥 ∈ 𝐶 ∧ ¬ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)))
8	3, 5, 7	3bitri 296	. . . . . 6 ⊢ (𝑥 ∈ ((𝐴 ∖ 𝐵) ∩ 𝐶) ↔ (𝑥 ∈ 𝐶 ∧ ¬ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)))
9	2, 8	xchbinxr 334	. . . . 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 234	. . . 4 ⊢ (¬ 𝑥 ∈ ((𝐴 ∖ 𝐵) ∩ 𝐶) → ((𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐵)))
12	11	al2imi 1817	. . 3 ⊢ (∀𝑥 ¬ 𝑥 ∈ ((𝐴 ∖ 𝐵) ∩ 𝐶) → (∀𝑥(𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐴) → ∀𝑥(𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐵)))
13		dfss2 3967	. . 3 ⊢ (𝐶 ⊆ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐴))
14		dfss2 3967	. . 3 ⊢ (𝐶 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐵))
15	12, 13, 14	3imtr4g 295	. 2 ⊢ (∀𝑥 ¬ 𝑥 ∈ ((𝐴 ∖ 𝐵) ∩ 𝐶) → (𝐶 ⊆ 𝐴 → 𝐶 ⊆ 𝐵))
16	1, 15	sylbi 216	1 ⊢ (((𝐴 ∖ 𝐵) ∩ 𝐶) = ∅ → (𝐶 ⊆ 𝐴 → 𝐶 ⊆ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396  ∀wal 1539   = wceq 1541   ∈ wcel 2106   ∖ cdif 3944   ∩ cin 3946   ⊆ wss 3947  ∅c0 4321

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-dif 3950  df-in 3954  df-ss 3964  df-nul 4322

This theorem is referenced by:  tz7.7  6387  tfi  7838  lebnumlem3  24470