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Mirrors > Home > MPE Home > Th. List > elopab | Structured version Visualization version GIF version |
Description: Membership in a class abstraction of 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 3212 | . 2 ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} → 𝐴 ∈ V) | |
2 | opex 4932 | . . . . 5 ⊢ 〈𝑥, 𝑦〉 ∈ V | |
3 | eleq1 2689 | . . . . 5 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (𝐴 ∈ V ↔ 〈𝑥, 𝑦〉 ∈ V)) | |
4 | 2, 3 | mpbiri 248 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → 𝐴 ∈ V) |
5 | 4 | adantr 481 | . . 3 ⊢ ((𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜑) → 𝐴 ∈ V) |
6 | 5 | exlimivv 1860 | . 2 ⊢ (∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜑) → 𝐴 ∈ V) |
7 | eqeq1 2626 | . . . . 5 ⊢ (𝑧 = 𝐴 → (𝑧 = 〈𝑥, 𝑦〉 ↔ 𝐴 = 〈𝑥, 𝑦〉)) | |
8 | 7 | anbi1d 741 | . . . 4 ⊢ (𝑧 = 𝐴 → ((𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜑))) |
9 | 8 | 2exbidv 1852 | . . 3 ⊢ (𝑧 = 𝐴 → (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜑))) |
10 | df-opab 4713 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
11 | 9, 10 | elab2g 3353 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜑))) |
12 | 1, 6, 11 | pm5.21nii 368 | 1 ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 Vcvv 3200 〈cop 4183 {copab 4712 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-opab 4713 |
This theorem is referenced by: opelopabsbALT 4984 opelopabsb 4985 opelopabt 4987 opelopabga 4988 opabn0 5006 iunopab 5012 elopabr 5013 epelg 5030 elxp 5131 elopaelxp 5191 elopaba 5232 elcnv 5299 dfmpt3 6014 fmptsng 6434 fmptsnd 6435 0neqopab 6698 opabex3d 7145 opabex3 7146 fsplit 7282 rtrclreclem3 13800 isfunc 16524 griedg0ssusgr 26157 rgrusgrprc 26485 brabgaf 29420 qqhval2 30026 eulerpartlemgvv 30438 poimirlem26 33435 dicelval3 36469 pellexlem5 37397 pellex 37399 opelopab4 38767 sprsymrelfvlem 41740 uspgrsprf 41754 uspgrsprf1 41755 |
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