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Mirrors > Home > ILE Home > Th. List > unissel | GIF version |
Description: Condition turning a subclass relationship for union into an equality. (Contributed by NM, 18-Jul-2006.) |
Ref | Expression |
---|---|
unissel | ⊢ ((∪ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐴) → ∪ 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 107 | . 2 ⊢ ((∪ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐴) → ∪ 𝐴 ⊆ 𝐵) | |
2 | elssuni 3629 | . . 3 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝐴) | |
3 | 2 | adantl 271 | . 2 ⊢ ((∪ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊆ ∪ 𝐴) |
4 | 1, 3 | eqssd 3016 | 1 ⊢ ((∪ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐴) → ∪ 𝐴 = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1284 ∈ wcel 1433 ⊆ wss 2973 ∪ cuni 3601 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-in 2979 df-ss 2986 df-uni 3602 |
This theorem is referenced by: elpwuni 3762 |
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