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Mirrors > Home > MPE Home > Th. List > unissel | Structured version Visualization version 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 483 | . 2 ⊢ ((∪ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐴) → ∪ 𝐴 ⊆ 𝐵) | |
2 | elssuni 4898 | . . 3 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝐴) | |
3 | 2 | adantl 482 | . 2 ⊢ ((∪ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊆ ∪ 𝐴) |
4 | 1, 3 | eqssd 3961 | 1 ⊢ ((∪ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐴) → ∪ 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ⊆ wss 3910 ∪ cuni 4865 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-v 3447 df-in 3917 df-ss 3927 df-uni 4866 |
This theorem is referenced by: elpwuni 5065 mretopd 22441 toponmre 22442 neiptopuni 22479 filunibas 23230 unidmvol 24903 unicls 32424 carsguni 32848 onintunirab 41538 |
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