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Mirrors > Home > MPE Home > Th. List > elpwuni | Structured version Visualization version GIF version |
Description: Relationship for power class and union. (Contributed by NM, 18-Jul-2006.) |
Ref | Expression |
---|---|
elpwuni | ⊢ (𝐵 ∈ 𝐴 → (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sspwuni 4994 | . 2 ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵) | |
2 | unissel 4838 | . . . 4 ⊢ ((∪ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐴) → ∪ 𝐴 = 𝐵) | |
3 | 2 | expcom 417 | . . 3 ⊢ (𝐵 ∈ 𝐴 → (∪ 𝐴 ⊆ 𝐵 → ∪ 𝐴 = 𝐵)) |
4 | eqimss 3943 | . . 3 ⊢ (∪ 𝐴 = 𝐵 → ∪ 𝐴 ⊆ 𝐵) | |
5 | 3, 4 | impbid1 228 | . 2 ⊢ (𝐵 ∈ 𝐴 → (∪ 𝐴 ⊆ 𝐵 ↔ ∪ 𝐴 = 𝐵)) |
6 | 1, 5 | syl5bb 286 | 1 ⊢ (𝐵 ∈ 𝐴 → (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∈ wcel 2112 ⊆ wss 3853 𝒫 cpw 4499 ∪ cuni 4805 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-11 2160 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-v 3400 df-in 3860 df-ss 3870 df-pw 4501 df-uni 4806 |
This theorem is referenced by: mreuni 17057 ustuni 23078 utopbas 23087 issgon 31757 br2base 31902 |
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