Theorem inunissunidif 36746

Description: Theorem about subsets of the difference of unions. (Contributed by ML, 29-Mar-2021.)

Assertion

Ref	Expression
inunissunidif	⊢ ((𝐴 ∩ ∪ 𝐶) = ∅ → (𝐴 ⊆ ∪ 𝐵 ↔ 𝐴 ⊆ ∪ (𝐵 ∖ 𝐶)))

Proof of Theorem inunissunidif

Step	Hyp	Ref	Expression
1		reldisj 4443	. . . 4 ⊢ (𝐴 ⊆ ∪ 𝐵 → ((𝐴 ∩ ∪ 𝐶) = ∅ ↔ 𝐴 ⊆ (∪ 𝐵 ∖ ∪ 𝐶)))
2		difunieq 36745	. . . . 5 ⊢ (∪ 𝐵 ∖ ∪ 𝐶) ⊆ ∪ (𝐵 ∖ 𝐶)
3		sstr 3982	. . . . 5 ⊢ ((𝐴 ⊆ (∪ 𝐵 ∖ ∪ 𝐶) ∧ (∪ 𝐵 ∖ ∪ 𝐶) ⊆ ∪ (𝐵 ∖ 𝐶)) → 𝐴 ⊆ ∪ (𝐵 ∖ 𝐶))
4	2, 3	mpan2 688	. . . 4 ⊢ (𝐴 ⊆ (∪ 𝐵 ∖ ∪ 𝐶) → 𝐴 ⊆ ∪ (𝐵 ∖ 𝐶))
5	1, 4	syl6bi 253	. . 3 ⊢ (𝐴 ⊆ ∪ 𝐵 → ((𝐴 ∩ ∪ 𝐶) = ∅ → 𝐴 ⊆ ∪ (𝐵 ∖ 𝐶)))
6	5	com12 32	. 2 ⊢ ((𝐴 ∩ ∪ 𝐶) = ∅ → (𝐴 ⊆ ∪ 𝐵 → 𝐴 ⊆ ∪ (𝐵 ∖ 𝐶)))
7		difss 4123	. . . 4 ⊢ (𝐵 ∖ 𝐶) ⊆ 𝐵
8	7	unissi 4908	. . 3 ⊢ ∪ (𝐵 ∖ 𝐶) ⊆ ∪ 𝐵
9		sstr 3982	. . 3 ⊢ ((𝐴 ⊆ ∪ (𝐵 ∖ 𝐶) ∧ ∪ (𝐵 ∖ 𝐶) ⊆ ∪ 𝐵) → 𝐴 ⊆ ∪ 𝐵)
10	8, 9	mpan2 688	. 2 ⊢ (𝐴 ⊆ ∪ (𝐵 ∖ 𝐶) → 𝐴 ⊆ ∪ 𝐵)
11	6, 10	impbid1 224	1 ⊢ ((𝐴 ∩ ∪ 𝐶) = ∅ → (𝐴 ⊆ ∪ 𝐵 ↔ 𝐴 ⊆ ∪ (𝐵 ∖ 𝐶)))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1533   ∖ cdif 3937   ∩ cin 3939   ⊆ wss 3940  ∅c0 4314  ∪ cuni 4899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-v 3468  df-dif 3943  df-in 3947  df-ss 3957  df-nul 4315  df-uni 4900

This theorem is referenced by:  pibt2  36788