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Mirrors > Home > ILE Home > Th. List > ifelpwung | GIF version |
Description: Existence of a conditional class, quantitative version (closed form). (Contributed by BJ, 15-Aug-2024.) |
Ref | Expression |
---|---|
ifelpwung | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → if(𝜑, 𝐴, 𝐵) ∈ 𝒫 (𝐴 ∪ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifssun 3539 | . 2 ⊢ if(𝜑, 𝐴, 𝐵) ⊆ (𝐴 ∪ 𝐵) | |
2 | unexg 4426 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V) | |
3 | elpw2g 4140 | . . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ V → (if(𝜑, 𝐴, 𝐵) ∈ 𝒫 (𝐴 ∪ 𝐵) ↔ if(𝜑, 𝐴, 𝐵) ⊆ (𝐴 ∪ 𝐵))) | |
4 | 2, 3 | syl 14 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (if(𝜑, 𝐴, 𝐵) ∈ 𝒫 (𝐴 ∪ 𝐵) ↔ if(𝜑, 𝐴, 𝐵) ⊆ (𝐴 ∪ 𝐵))) |
5 | 1, 4 | mpbiri 167 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → if(𝜑, 𝐴, 𝐵) ∈ 𝒫 (𝐴 ∪ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 2141 Vcvv 2730 ∪ cun 3119 ⊆ wss 3121 ifcif 3525 𝒫 cpw 3564 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pr 4192 ax-un 4416 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-rex 2454 df-rab 2457 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-if 3526 df-pw 3566 df-sn 3587 df-pr 3588 df-uni 3795 |
This theorem is referenced by: ifelpwund 4465 ifelpwun 4466 |
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