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Mirrors > Home > MPE Home > Th. List > opabn0 | Structured version Visualization version GIF version |
Description: Nonempty ordered pair class abstraction. (Contributed by NM, 10-Oct-2007.) |
Ref | Expression |
---|---|
opabn0 | ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥∃𝑦𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0 3890 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ≠ ∅ ↔ ∃𝑧 𝑧 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
2 | elopab 4908 | . . . 4 ⊢ (𝑧 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
3 | 2 | exbii 1764 | . . 3 ⊢ (∃𝑧 𝑧 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑧∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)) |
4 | exrot3 2032 | . . . 4 ⊢ (∃𝑧∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦∃𝑧(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
5 | opex 4859 | . . . . . . 7 ⊢ 〈𝑥, 𝑦〉 ∈ V | |
6 | 5 | isseti 3182 | . . . . . 6 ⊢ ∃𝑧 𝑧 = 〈𝑥, 𝑦〉 |
7 | 19.41v 1901 | . . . . . 6 ⊢ (∃𝑧(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (∃𝑧 𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
8 | 6, 7 | mpbiran 955 | . . . . 5 ⊢ (∃𝑧(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ 𝜑) |
9 | 8 | 2exbii 1765 | . . . 4 ⊢ (∃𝑥∃𝑦∃𝑧(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦𝜑) |
10 | 4, 9 | bitri 263 | . . 3 ⊢ (∃𝑧∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦𝜑) |
11 | 3, 10 | bitri 263 | . 2 ⊢ (∃𝑧 𝑧 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦𝜑) |
12 | 1, 11 | bitri 263 | 1 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥∃𝑦𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ≠ wne 2780 ∅c0 3874 〈cop 4131 {copab 4642 |
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-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 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-ne 2782 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-opab 4644 |
This theorem is referenced by: csbopab 4932 dvdsrval 18468 thlle 19860 bcthlem5 22933 lgsquadlem3 24907 |
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