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Mirrors > Home > MPE Home > Th. List > ssunieq | Structured version Visualization version GIF version |
Description: Relationship implying union. (Contributed by NM, 10-Nov-1999.) |
Ref | Expression |
---|---|
ssunieq | ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝑥 ⊆ 𝐴) → 𝐴 = ∪ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elssuni 4934 | . . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ⊆ ∪ 𝐵) | |
2 | unissb 4936 | . . . 4 ⊢ (∪ 𝐵 ⊆ 𝐴 ↔ ∀𝑥 ∈ 𝐵 𝑥 ⊆ 𝐴) | |
3 | 2 | biimpri 227 | . . 3 ⊢ (∀𝑥 ∈ 𝐵 𝑥 ⊆ 𝐴 → ∪ 𝐵 ⊆ 𝐴) |
4 | 1, 3 | anim12i 612 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝑥 ⊆ 𝐴) → (𝐴 ⊆ ∪ 𝐵 ∧ ∪ 𝐵 ⊆ 𝐴)) |
5 | eqss 3992 | . 2 ⊢ (𝐴 = ∪ 𝐵 ↔ (𝐴 ⊆ ∪ 𝐵 ∧ ∪ 𝐵 ⊆ 𝐴)) | |
6 | 4, 5 | sylibr 233 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝑥 ⊆ 𝐴) → 𝐴 = ∪ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3055 ⊆ wss 3943 ∪ cuni 4902 |
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-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-v 3470 df-in 3950 df-ss 3960 df-uni 4903 |
This theorem is referenced by: unimax 4941 shsspwh 31004 |
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