Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 4312 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ≠ ∅ ↔ ∃𝑧 𝑧 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
2 | elopab 5416 | . . . 4 ⊢ (𝑧 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
3 | 2 | exbii 1848 | . . 3 ⊢ (∃𝑧 𝑧 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑧∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)) |
4 | exrot3 2172 | . . . 4 ⊢ (∃𝑧∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦∃𝑧(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
5 | opex 5358 | . . . . . . 7 ⊢ 〈𝑥, 𝑦〉 ∈ V | |
6 | 5 | isseti 3510 | . . . . . 6 ⊢ ∃𝑧 𝑧 = 〈𝑥, 𝑦〉 |
7 | 19.41v 1950 | . . . . . 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 1849 | . . . 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 1537 ∃wex 1780 ∈ wcel 2114 ≠ wne 3018 ∅c0 4293 〈cop 4575 {copab 5130 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-opab 5131 |
This theorem is referenced by: opab0 5443 csbopab 5444 dvdsrval 19397 thlle 20843 bcthlem5 23933 lgsquadlem3 25960 br1cosscnvxrn 35716 |
Copyright terms: Public domain | W3C validator |