Theorem inunissunidif 34659

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 4402	. . . 4 ⊢ (𝐴 ⊆ ∪ 𝐵 → ((𝐴 ∩ ∪ 𝐶) = ∅ ↔ 𝐴 ⊆ (∪ 𝐵 ∖ ∪ 𝐶)))
2		difunieq 34658	. . . . 5 ⊢ (∪ 𝐵 ∖ ∪ 𝐶) ⊆ ∪ (𝐵 ∖ 𝐶)
3		sstr 3975	. . . . 5 ⊢ ((𝐴 ⊆ (∪ 𝐵 ∖ ∪ 𝐶) ∧ (∪ 𝐵 ∖ ∪ 𝐶) ⊆ ∪ (𝐵 ∖ 𝐶)) → 𝐴 ⊆ ∪ (𝐵 ∖ 𝐶))
4	2, 3	mpan2 689	. . . 4 ⊢ (𝐴 ⊆ (∪ 𝐵 ∖ ∪ 𝐶) → 𝐴 ⊆ ∪ (𝐵 ∖ 𝐶))
5	1, 4	syl6bi 255	. . 3 ⊢ (𝐴 ⊆ ∪ 𝐵 → ((𝐴 ∩ ∪ 𝐶) = ∅ → 𝐴 ⊆ ∪ (𝐵 ∖ 𝐶)))
6	5	com12 32	. 2 ⊢ ((𝐴 ∩ ∪ 𝐶) = ∅ → (𝐴 ⊆ ∪ 𝐵 → 𝐴 ⊆ ∪ (𝐵 ∖ 𝐶)))
7		difss 4108	. . . 4 ⊢ (𝐵 ∖ 𝐶) ⊆ 𝐵
8	7	unissi 4847	. . 3 ⊢ ∪ (𝐵 ∖ 𝐶) ⊆ ∪ 𝐵
9		sstr 3975	. . 3 ⊢ ((𝐴 ⊆ ∪ (𝐵 ∖ 𝐶) ∧ ∪ (𝐵 ∖ 𝐶) ⊆ ∪ 𝐵) → 𝐴 ⊆ ∪ 𝐵)
10	8, 9	mpan2 689	. 2 ⊢ (𝐴 ⊆ ∪ (𝐵 ∖ 𝐶) → 𝐴 ⊆ ∪ 𝐵)
11	6, 10	impbid1 227	1 ⊢ ((𝐴 ∩ ∪ 𝐶) = ∅ → (𝐴 ⊆ ∪ 𝐵 ↔ 𝐴 ⊆ ∪ (𝐵 ∖ 𝐶)))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1537   ∖ cdif 3933   ∩ cin 3935   ⊆ wss 3936  ∅c0 4291  ∪ cuni 4838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-v 3496  df-dif 3939  df-in 3943  df-ss 3952  df-nul 4292  df-uni 4839

This theorem is referenced by:  pibt2  34701