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Mirrors > Home > ILE Home > Th. List > elpwuni | 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 3861 | . 2 ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵) | |
2 | unissel 3729 | . . . 4 ⊢ ((∪ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐴) → ∪ 𝐴 = 𝐵) | |
3 | 2 | expcom 115 | . . 3 ⊢ (𝐵 ∈ 𝐴 → (∪ 𝐴 ⊆ 𝐵 → ∪ 𝐴 = 𝐵)) |
4 | eqimss 3115 | . . 3 ⊢ (∪ 𝐴 = 𝐵 → ∪ 𝐴 ⊆ 𝐵) | |
5 | 3, 4 | impbid1 141 | . 2 ⊢ (𝐵 ∈ 𝐴 → (∪ 𝐴 ⊆ 𝐵 ↔ ∪ 𝐴 = 𝐵)) |
6 | 1, 5 | syl5bb 191 | 1 ⊢ (𝐵 ∈ 𝐴 → (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 = 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1312 ∈ wcel 1461 ⊆ wss 3035 𝒫 cpw 3474 ∪ cuni 3700 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 |
This theorem depends on definitions: df-bi 116 df-tru 1315 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ral 2393 df-v 2657 df-in 3041 df-ss 3048 df-pw 3476 df-uni 3701 |
This theorem is referenced by: (None) |
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