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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 3983 | . 2 ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵) | |
2 | unissel 3850 | . . . 4 ⊢ ((∪ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐴) → ∪ 𝐴 = 𝐵) | |
3 | 2 | expcom 116 | . . 3 ⊢ (𝐵 ∈ 𝐴 → (∪ 𝐴 ⊆ 𝐵 → ∪ 𝐴 = 𝐵)) |
4 | eqimss 3221 | . . 3 ⊢ (∪ 𝐴 = 𝐵 → ∪ 𝐴 ⊆ 𝐵) | |
5 | 3, 4 | impbid1 142 | . 2 ⊢ (𝐵 ∈ 𝐴 → (∪ 𝐴 ⊆ 𝐵 ↔ ∪ 𝐴 = 𝐵)) |
6 | 1, 5 | bitrid 192 | 1 ⊢ (𝐵 ∈ 𝐴 → (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 = 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1363 ∈ wcel 2158 ⊆ wss 3141 𝒫 cpw 3587 ∪ cuni 3821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-v 2751 df-in 3147 df-ss 3154 df-pw 3589 df-uni 3822 |
This theorem is referenced by: (None) |
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