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Mirrors > Home > MPE Home > Th. List > ifpr | Structured version Visualization version GIF version |
Description: Membership of a conditional operator in an unordered pair. (Contributed by NM, 17-Jun-2007.) |
Ref | Expression |
---|---|
ifpr | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → if(𝜑, 𝐴, 𝐵) ∈ {𝐴, 𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3352 | . 2 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ V) | |
2 | elex 3352 | . 2 ⊢ (𝐵 ∈ 𝐷 → 𝐵 ∈ V) | |
3 | ifcl 4274 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → if(𝜑, 𝐴, 𝐵) ∈ V) | |
4 | ifeqor 4276 | . . . 4 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 ∨ if(𝜑, 𝐴, 𝐵) = 𝐵) | |
5 | elprg 4341 | . . . 4 ⊢ (if(𝜑, 𝐴, 𝐵) ∈ V → (if(𝜑, 𝐴, 𝐵) ∈ {𝐴, 𝐵} ↔ (if(𝜑, 𝐴, 𝐵) = 𝐴 ∨ if(𝜑, 𝐴, 𝐵) = 𝐵))) | |
6 | 4, 5 | mpbiri 248 | . . 3 ⊢ (if(𝜑, 𝐴, 𝐵) ∈ V → if(𝜑, 𝐴, 𝐵) ∈ {𝐴, 𝐵}) |
7 | 3, 6 | syl 17 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → if(𝜑, 𝐴, 𝐵) ∈ {𝐴, 𝐵}) |
8 | 1, 2, 7 | syl2an 495 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → if(𝜑, 𝐴, 𝐵) ∈ {𝐴, 𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 382 ∧ wa 383 = wceq 1632 ∈ wcel 2139 Vcvv 3340 ifcif 4230 {cpr 4323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-v 3342 df-un 3720 df-if 4231 df-sn 4322 df-pr 4324 |
This theorem is referenced by: suppr 8544 infpr 8576 uvcvvcl 20348 indf 30407 |
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