Theorem inunissunidif 37357

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 4458	. . . 4 ⊢ (𝐴 ⊆ ∪ 𝐵 → ((𝐴 ∩ ∪ 𝐶) = ∅ ↔ 𝐴 ⊆ (∪ 𝐵 ∖ ∪ 𝐶)))
2		difunieq 37356	. . . . 5 ⊢ (∪ 𝐵 ∖ ∪ 𝐶) ⊆ ∪ (𝐵 ∖ 𝐶)
3		sstr 4003	. . . . 5 ⊢ ((𝐴 ⊆ (∪ 𝐵 ∖ ∪ 𝐶) ∧ (∪ 𝐵 ∖ ∪ 𝐶) ⊆ ∪ (𝐵 ∖ 𝐶)) → 𝐴 ⊆ ∪ (𝐵 ∖ 𝐶))
4	2, 3	mpan2 691	. . . 4 ⊢ (𝐴 ⊆ (∪ 𝐵 ∖ ∪ 𝐶) → 𝐴 ⊆ ∪ (𝐵 ∖ 𝐶))
5	1, 4	biimtrdi 253	. . 3 ⊢ (𝐴 ⊆ ∪ 𝐵 → ((𝐴 ∩ ∪ 𝐶) = ∅ → 𝐴 ⊆ ∪ (𝐵 ∖ 𝐶)))
6	5	com12 32	. 2 ⊢ ((𝐴 ∩ ∪ 𝐶) = ∅ → (𝐴 ⊆ ∪ 𝐵 → 𝐴 ⊆ ∪ (𝐵 ∖ 𝐶)))
7		difss 4145	. . . 4 ⊢ (𝐵 ∖ 𝐶) ⊆ 𝐵
8	7	unissi 4920	. . 3 ⊢ ∪ (𝐵 ∖ 𝐶) ⊆ ∪ 𝐵
9		sstr 4003	. . 3 ⊢ ((𝐴 ⊆ ∪ (𝐵 ∖ 𝐶) ∧ ∪ (𝐵 ∖ 𝐶) ⊆ ∪ 𝐵) → 𝐴 ⊆ ∪ 𝐵)
10	8, 9	mpan2 691	. 2 ⊢ (𝐴 ⊆ ∪ (𝐵 ∖ 𝐶) → 𝐴 ⊆ ∪ 𝐵)
11	6, 10	impbid1 225	1 ⊢ ((𝐴 ∩ ∪ 𝐶) = ∅ → (𝐴 ⊆ ∪ 𝐵 ↔ 𝐴 ⊆ ∪ (𝐵 ∖ 𝐶)))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1536   ∖ cdif 3959   ∩ cin 3961   ⊆ wss 3962  ∅c0 4338  ∪ cuni 4911

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-12 2174  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-v 3479  df-dif 3965  df-in 3969  df-ss 3979  df-nul 4339  df-uni 4912

This theorem is referenced by:  pibt2  37399