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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 3493 | . 2 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝐴 ∈ V) | |
2 | opex 5465 | . . . . 5 ⊢ ⟨𝑥, 𝑦⟩ ∈ V | |
3 | eleq1 2822 | . . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ V ↔ ⟨𝑥, 𝑦⟩ ∈ V)) | |
4 | 2, 3 | mpbiri 258 | . . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴 ∈ V) |
5 | 4 | adantr 482 | . . 3 ⊢ ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝐴 ∈ V) |
6 | 5 | exlimivv 1936 | . 2 ⊢ (∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝐴 ∈ V) |
7 | elopabw 5527 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))) | |
8 | 1, 6, 7 | pm5.21nii 380 | 1 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 Vcvv 3475 ⟨cop 4635 {copab 5211 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-opab 5212 |
This theorem is referenced by: rexopabb 5529 vopelopabsb 5530 opelopabsb 5531 opelopabt 5533 opelopabga 5534 opabn0 5554 iunopabOLD 5561 elopabrOLD 5564 0nelopab 5568 0nelopabOLD 5569 elxp 5700 elopaelxpOLD 5767 elopaba 5809 elcnv 5877 dfmpt3 6685 fmptsng 7166 fmptsnd 7167 opabex3d 7952 opabex3rd 7953 opabex3 7954 fsplit 8103 rtrclreclem3 15007 isfunc 17814 griedg0ssusgr 28522 rgrusgrprc 28846 brabgaf 31837 qqhval2 32962 eulerpartlemgvv 33375 satfvsucsuc 34356 satf0op 34368 opelopabd 36022 opelopabb 36023 poimirlem26 36514 ecxrn 37257 dicelval3 40051 pellexlem5 41571 pellex 41573 opelopab4 43312 sprsymrelfvlem 46158 uspgrsprf 46524 uspgrsprf1 46525 |
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