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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 4230 | . . 3 ⊢ (𝑧 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
2 | 1 | exbii 1592 | . 2 ⊢ (∃𝑧 𝑧 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑧∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)) |
3 | exrot3 1677 | . 2 ⊢ (∃𝑧∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦∃𝑧(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
4 | vex 2724 | . . . . . 6 ⊢ 𝑥 ∈ V | |
5 | vex 2724 | . . . . . 6 ⊢ 𝑦 ∈ V | |
6 | 4, 5 | opex 4201 | . . . . 5 ⊢ 〈𝑥, 𝑦〉 ∈ V |
7 | 6 | isseti 2729 | . . . 4 ⊢ ∃𝑧 𝑧 = 〈𝑥, 𝑦〉 |
8 | 19.41v 1889 | . . . 4 ⊢ (∃𝑧(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (∃𝑧 𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
9 | 7, 8 | mpbiran 929 | . . 3 ⊢ (∃𝑧(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ 𝜑) |
10 | 9 | 2exbii 1593 | . 2 ⊢ (∃𝑥∃𝑦∃𝑧(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦𝜑) |
11 | 2, 3, 10 | 3bitri 205 | 1 ⊢ (∃𝑧 𝑧 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦𝜑) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1342 ∃wex 1479 ∈ wcel 2135 〈cop 3573 {copab 4036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-opab 4038 |
This theorem is referenced by: (None) |
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