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Mirrors > Home > MPE Home > Th. List > Mathboxes > inunissunidif | Structured version Visualization version GIF version |
Description: Theorem about subsets of the difference of unions. (Contributed by ML, 29-Mar-2021.) |
Ref | Expression |
---|---|
inunissunidif | ⊢ ((𝐴 ∩ ∪ 𝐶) = ∅ → (𝐴 ⊆ ∪ 𝐵 ↔ 𝐴 ⊆ ∪ (𝐵 ∖ 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reldisj 4391 | . . . 4 ⊢ (𝐴 ⊆ ∪ 𝐵 → ((𝐴 ∩ ∪ 𝐶) = ∅ ↔ 𝐴 ⊆ (∪ 𝐵 ∖ ∪ 𝐶))) | |
2 | difunieq 35541 | . . . . 5 ⊢ (∪ 𝐵 ∖ ∪ 𝐶) ⊆ ∪ (𝐵 ∖ 𝐶) | |
3 | sstr 3934 | . . . . 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 252 | . . 3 ⊢ (𝐴 ⊆ ∪ 𝐵 → ((𝐴 ∩ ∪ 𝐶) = ∅ → 𝐴 ⊆ ∪ (𝐵 ∖ 𝐶))) |
6 | 5 | com12 32 | . 2 ⊢ ((𝐴 ∩ ∪ 𝐶) = ∅ → (𝐴 ⊆ ∪ 𝐵 → 𝐴 ⊆ ∪ (𝐵 ∖ 𝐶))) |
7 | difss 4071 | . . . 4 ⊢ (𝐵 ∖ 𝐶) ⊆ 𝐵 | |
8 | 7 | unissi 4854 | . . 3 ⊢ ∪ (𝐵 ∖ 𝐶) ⊆ ∪ 𝐵 |
9 | sstr 3934 | . . 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))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∖ cdif 3889 ∩ cin 3891 ⊆ wss 3892 ∅c0 4262 ∪ cuni 4845 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-12 2175 ax-ext 2711 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-clab 2718 df-cleq 2732 df-clel 2818 df-ral 3071 df-v 3433 df-dif 3895 df-in 3899 df-ss 3909 df-nul 4263 df-uni 4846 |
This theorem is referenced by: pibt2 35584 |
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