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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 4303 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ≠ ∅ ↔ ∃𝑧 𝑧 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
2 | elopab 5407 | . . . 4 ⊢ (𝑧 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
3 | 2 | exbii 1847 | . . 3 ⊢ (∃𝑧 𝑧 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑧∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)) |
4 | exrot3 2171 | . . . 4 ⊢ (∃𝑧∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦∃𝑧(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
5 | opex 5349 | . . . . . . 7 ⊢ 〈𝑥, 𝑦〉 ∈ V | |
6 | 5 | isseti 3505 | . . . . . 6 ⊢ ∃𝑧 𝑧 = 〈𝑥, 𝑦〉 |
7 | 19.41v 1949 | . . . . . 6 ⊢ (∃𝑧(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (∃𝑧 𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
8 | 6, 7 | mpbiran 707 | . . . . 5 ⊢ (∃𝑧(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ 𝜑) |
9 | 8 | 2exbii 1848 | . . . 4 ⊢ (∃𝑥∃𝑦∃𝑧(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦𝜑) |
10 | 4, 9 | bitri 277 | . . 3 ⊢ (∃𝑧∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦𝜑) |
11 | 3, 10 | bitri 277 | . 2 ⊢ (∃𝑧 𝑧 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦𝜑) |
12 | 1, 11 | bitri 277 | 1 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥∃𝑦𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1536 ∃wex 1779 ∈ wcel 2113 ≠ wne 3015 ∅c0 4284 〈cop 4566 {copab 5121 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-opab 5122 |
This theorem is referenced by: opab0 5434 csbopab 5435 dvdsrval 19390 thlle 20836 bcthlem5 23926 lgsquadlem3 25956 br1cosscnvxrn 35747 |
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