Theorem difin0ss 4320

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 4296	. 2 ⊢ (((𝐴 ∖ 𝐵) ∩ 𝐶) = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ ((𝐴 ∖ 𝐵) ∩ 𝐶))
2		iman 407	. . . . . 6 ⊢ ((𝑥 ∈ 𝐶 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) ↔ ¬ (𝑥 ∈ 𝐶 ∧ ¬ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)))
3		elin 4092	. . . . . . 7 ⊢ (𝑥 ∈ ((𝐴 ∖ 𝐵) ∩ 𝐶) ↔ (𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥 ∈ 𝐶))
4		eldif 4048	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
5	4	anbi2ci 637	. . . . . . 7 ⊢ ((𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)))
6		annim 409	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ↔ ¬ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
7	6	anbi2i 635	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) ↔ (𝑥 ∈ 𝐶 ∧ ¬ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)))
8	3, 5, 7	3bitri 300	. . . . . 6 ⊢ (𝑥 ∈ ((𝐴 ∖ 𝐵) ∩ 𝐶) ↔ (𝑥 ∈ 𝐶 ∧ ¬ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)))
9	2, 8	xchbinxr 338	. . . . 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 238	. . . 4 ⊢ (¬ 𝑥 ∈ ((𝐴 ∖ 𝐵) ∩ 𝐶) → ((𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐵)))
12	11	al2imi 1847	. . 3 ⊢ (∀𝑥 ¬ 𝑥 ∈ ((𝐴 ∖ 𝐵) ∩ 𝐶) → (∀𝑥(𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐴) → ∀𝑥(𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐵)))
13		df-ss 3897	. . 3 ⊢ (𝐶 ⊆ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐴))
14		df-ss 3897	. . 3 ⊢ (𝐶 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐵))
15	12, 13, 14	3imtr4g 299	. 2 ⊢ (∀𝑥 ¬ 𝑥 ∈ ((𝐴 ∖ 𝐵) ∩ 𝐶) → (𝐶 ⊆ 𝐴 → 𝐶 ⊆ 𝐵))
16	1, 15	sylbi 220	1 ⊢ (((𝐴 ∖ 𝐵) ∩ 𝐶) = ∅ → (𝐶 ⊆ 𝐴 → 𝐶 ⊆ 𝐵))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401  ∀wal 1567   = wceq 1569   ∈ wcel 2144   ⊆ wss 3896   ∖ cdif 4044   ∩ cin 4088  ∅c0 4277

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-nul 4278

This theorem is used by:  tz7.7  6391  tfi  7868  lebnumlem3  25293