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Mirrors > Home > ILE Home > Th. List > ifelpwun | GIF version |
Description: Existence of a conditional class, quantitative version (inference form). (Contributed by BJ, 15-Aug-2024.) |
Ref | Expression |
---|---|
ifelpwun.1 | ⊢ 𝐴 ∈ V |
ifelpwun.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
ifelpwun | ⊢ if(𝜑, 𝐴, 𝐵) ∈ 𝒫 (𝐴 ∪ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifelpwun.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | ifelpwun.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | ifelpwung 4440 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → if(𝜑, 𝐴, 𝐵) ∈ 𝒫 (𝐴 ∪ 𝐵)) | |
4 | 1, 2, 3 | mp2an 423 | 1 ⊢ if(𝜑, 𝐴, 𝐵) ∈ 𝒫 (𝐴 ∪ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2128 Vcvv 2712 ∪ cun 3100 ifcif 3505 𝒫 cpw 3543 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pr 4169 ax-un 4393 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-rex 2441 df-rab 2444 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-uni 3773 |
This theorem is referenced by: fmelpw1o 13352 bj-charfun 13353 |
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