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Mirrors > Home > ILE Home > Th. List > elopab | GIF version |
Description: Membership in a class abstraction of ordered pairs. (Contributed by NM, 24-Mar-1998.) |
Ref | Expression |
---|---|
elopab | ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2748 | . 2 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝐴 ∈ V) | |
2 | vex 2740 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | vex 2740 | . . . . . 6 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | opex 4229 | . . . . 5 ⊢ ⟨𝑥, 𝑦⟩ ∈ V |
5 | eleq1 2240 | . . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ V ↔ ⟨𝑥, 𝑦⟩ ∈ V)) | |
6 | 4, 5 | mpbiri 168 | . . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴 ∈ V) |
7 | 6 | adantr 276 | . . 3 ⊢ ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝐴 ∈ V) |
8 | 7 | exlimivv 1896 | . 2 ⊢ (∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝐴 ∈ V) |
9 | eqeq1 2184 | . . . . 5 ⊢ (𝑧 = 𝐴 → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ 𝐴 = ⟨𝑥, 𝑦⟩)) | |
10 | 9 | anbi1d 465 | . . . 4 ⊢ (𝑧 = 𝐴 → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))) |
11 | 10 | 2exbidv 1868 | . . 3 ⊢ (𝑧 = 𝐴 → (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))) |
12 | df-opab 4065 | . . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} | |
13 | 11, 12 | elab2g 2884 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))) |
14 | 1, 8, 13 | pm5.21nii 704 | 1 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 = wceq 1353 ∃wex 1492 ∈ wcel 2148 Vcvv 2737 ⟨cop 3595 {copab 4063 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-opab 4065 |
This theorem is referenced by: opelopabsbALT 4259 opelopabsb 4260 opelopabt 4262 opelopabga 4263 opabm 4280 iunopab 4281 epelg 4290 elxp 4643 elco 4793 elcnv 4804 dfmpt3 5338 0neqopab 5919 brabvv 5920 opabex3d 6121 opabex3 6122 |
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