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Mirrors > Home > MPE Home > Th. List > elprg | Structured version Visualization version GIF version |
Description: A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15, generalized. (Contributed by NM, 13-Sep-1995.) |
Ref | Expression |
---|---|
elprg | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2614 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐵 ↔ 𝐴 = 𝐵)) | |
2 | eqeq1 2614 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐶 ↔ 𝐴 = 𝐶)) | |
3 | 1, 2 | orbi12d 742 | . 2 ⊢ (𝑥 = 𝐴 → ((𝑥 = 𝐵 ∨ 𝑥 = 𝐶) ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))) |
4 | dfpr2 4143 | . 2 ⊢ {𝐵, 𝐶} = {𝑥 ∣ (𝑥 = 𝐵 ∨ 𝑥 = 𝐶)} | |
5 | 3, 4 | elab2g 3322 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∨ wo 382 = wceq 1475 ∈ wcel 1977 {cpr 4127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-un 3545 df-sn 4126 df-pr 4128 |
This theorem is referenced by: elpri 4145 elpr 4146 elpr2 4147 elpr2OLD 4148 eldifpr 4152 eltpg 4174 ifpr 4180 prid1g 4239 ssprss 4296 preq1b 4317 ordunpr 6918 hashtpg 13121 cnsubrg 19625 atandm 24403 nbgra0nb 25958 eupath2lem1 26504 eliccioo 28970 nelpr2 38288 nelpr1 38289 sfprmdvdsmersenne 40058 1egrvtxdg0 40727 eupth2lem1 41386 |
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