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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 4774 | . . 3 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝐴) | |
3 | 2 | adantl 482 | . 2 ⊢ ((∪ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊆ ∪ 𝐴) |
4 | 1, 3 | eqssd 3906 | 1 ⊢ ((∪ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐴) → ∪ 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1522 ∈ wcel 2081 ⊆ wss 3859 ∪ cuni 4745 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-ext 2769 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-v 3439 df-in 3866 df-ss 3874 df-uni 4746 |
This theorem is referenced by: elpwuni 4926 mretopd 21384 toponmre 21385 neiptopuni 21422 filunibas 22173 unidmvol 23825 unicls 30763 carsguni 31183 |
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