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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 468 | . 2 ⊢ ((∪ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐴) → ∪ 𝐴 ⊆ 𝐵) | |
2 | elssuni 4603 | . . 3 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝐴) | |
3 | 2 | adantl 467 | . 2 ⊢ ((∪ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊆ ∪ 𝐴) |
4 | 1, 3 | eqssd 3769 | 1 ⊢ ((∪ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐴) → ∪ 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ⊆ wss 3723 ∪ cuni 4574 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-v 3353 df-in 3730 df-ss 3737 df-uni 4575 |
This theorem is referenced by: elpwuni 4750 mretopd 21113 toponmre 21114 neiptopuni 21151 filunibas 21901 unidmvol 23525 unicls 30285 carsguni 30706 |
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