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Mirrors > Home > ILE Home > Th. List > opabm | GIF version |
Description: Inhabited ordered pair class abstraction. (Contributed by Jim Kingdon, 29-Sep-2018.) |
Ref | Expression |
---|---|
opabm | ⊢ (∃𝑧 𝑧 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elopab 4254 | . . 3 ⊢ (𝑧 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
2 | 1 | exbii 1605 | . 2 ⊢ (∃𝑧 𝑧 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑧∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)) |
3 | exrot3 1690 | . 2 ⊢ (∃𝑧∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦∃𝑧(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
4 | vex 2740 | . . . . . 6 ⊢ 𝑥 ∈ V | |
5 | vex 2740 | . . . . . 6 ⊢ 𝑦 ∈ V | |
6 | 4, 5 | opex 4225 | . . . . 5 ⊢ 〈𝑥, 𝑦〉 ∈ V |
7 | 6 | isseti 2745 | . . . 4 ⊢ ∃𝑧 𝑧 = 〈𝑥, 𝑦〉 |
8 | 19.41v 1902 | . . . 4 ⊢ (∃𝑧(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (∃𝑧 𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
9 | 7, 8 | mpbiran 940 | . . 3 ⊢ (∃𝑧(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ 𝜑) |
10 | 9 | 2exbii 1606 | . 2 ⊢ (∃𝑥∃𝑦∃𝑧(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦𝜑) |
11 | 2, 3, 10 | 3bitri 206 | 1 ⊢ (∃𝑧 𝑧 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦𝜑) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 = wceq 1353 ∃wex 1492 ∈ wcel 2148 〈cop 3594 {copab 4060 |
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 4118 ax-pow 4171 ax-pr 4205 |
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 3576 df-sn 3597 df-pr 3598 df-op 3600 df-opab 4062 |
This theorem is referenced by: (None) |
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