Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 3510 | . 2 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ V) | |
2 | elex 3510 | . 2 ⊢ (𝐵 ∈ 𝐷 → 𝐵 ∈ V) | |
3 | ifcl 4507 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → if(𝜑, 𝐴, 𝐵) ∈ V) | |
4 | ifeqor 4512 | . . . 4 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 ∨ if(𝜑, 𝐴, 𝐵) = 𝐵) | |
5 | elprg 4578 | . . . 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 259 | . . 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 595 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → if(𝜑, 𝐴, 𝐵) ∈ {𝐴, 𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∨ wo 841 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ifcif 4463 {cpr 4559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-v 3494 df-un 3938 df-if 4464 df-sn 4558 df-pr 4560 |
This theorem is referenced by: suppr 8923 infpr 8955 uvcvvcl 20859 indf 31173 |
Copyright terms: Public domain | W3C validator |