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| Mirrors > Home > MPE Home > Th. List > elopab | Structured version Visualization version GIF version | ||
| Description: Membership in a class abstraction of ordered pairs. (Contributed by NM, 24-Mar-1998.) |
| Ref | Expression |
|---|---|
| elopab | ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜑)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elex 3501 | . 2 ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} → 𝐴 ∈ V) | |
| 2 | opex 5469 | . . . . 5 ⊢ 〈𝑥, 𝑦〉 ∈ V | |
| 3 | eleq1 2829 | . . . . 5 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (𝐴 ∈ V ↔ 〈𝑥, 𝑦〉 ∈ V)) | |
| 4 | 2, 3 | mpbiri 258 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → 𝐴 ∈ V) |
| 5 | 4 | adantr 480 | . . 3 ⊢ ((𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜑) → 𝐴 ∈ V) |
| 6 | 5 | exlimivv 1932 | . 2 ⊢ (∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜑) → 𝐴 ∈ V) |
| 7 | elopabw 5531 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜑))) | |
| 8 | 1, 6, 7 | pm5.21nii 378 | 1 ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜑)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 Vcvv 3480 〈cop 4632 {copab 5205 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-opab 5206 |
| This theorem is referenced by: rexopabb 5533 vopelopabsb 5534 opelopabsb 5535 opelopabt 5537 opelopabga 5538 opabn0 5558 iunopabOLD 5565 elopabrOLD 5568 0nelopab 5572 elxp 5708 elopaelxpOLD 5776 elopaba 5818 elcnv 5887 cnvopab 6157 dfmpt3 6702 fmptsng 7188 fmptsnd 7189 opabex3d 7990 opabex3rd 7991 opabex3 7992 fsplit 8142 rtrclreclem3 15099 isfunc 17909 griedg0ssusgr 29282 rgrusgrprc 29607 brab2d 32619 brabgaf 32620 qqhval2 33983 eulerpartlemgvv 34378 satfvsucsuc 35370 satf0op 35382 opelopabd 37142 opelopabb 37143 poimirlem26 37653 ecxrn 38388 dicelval3 41182 pellexlem5 42844 pellex 42846 opelopab4 44571 sprsymrelfvlem 47477 uspgrsprf 48062 uspgrsprf1 48063 brab2dd 48741 |
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