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Mirrors > Home > ILE Home > Th. List > elopab | 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 2723 | . 2 ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} → 𝐴 ∈ V) | |
2 | vex 2715 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | vex 2715 | . . . . . 6 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | opex 4190 | . . . . 5 ⊢ 〈𝑥, 𝑦〉 ∈ V |
5 | eleq1 2220 | . . . . 5 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (𝐴 ∈ V ↔ 〈𝑥, 𝑦〉 ∈ V)) | |
6 | 4, 5 | mpbiri 167 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → 𝐴 ∈ V) |
7 | 6 | adantr 274 | . . 3 ⊢ ((𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜑) → 𝐴 ∈ V) |
8 | 7 | exlimivv 1876 | . 2 ⊢ (∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜑) → 𝐴 ∈ V) |
9 | eqeq1 2164 | . . . . 5 ⊢ (𝑧 = 𝐴 → (𝑧 = 〈𝑥, 𝑦〉 ↔ 𝐴 = 〈𝑥, 𝑦〉)) | |
10 | 9 | anbi1d 461 | . . . 4 ⊢ (𝑧 = 𝐴 → ((𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜑))) |
11 | 10 | 2exbidv 1848 | . . 3 ⊢ (𝑧 = 𝐴 → (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜑))) |
12 | df-opab 4027 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
13 | 11, 12 | elab2g 2859 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜑))) |
14 | 1, 8, 13 | pm5.21nii 694 | 1 ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1335 ∃wex 1472 ∈ wcel 2128 Vcvv 2712 〈cop 3563 {copab 4025 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-pow 4136 ax-pr 4170 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-opab 4027 |
This theorem is referenced by: opelopabsbALT 4220 opelopabsb 4221 opelopabt 4223 opelopabga 4224 opabm 4241 iunopab 4242 epelg 4251 elxp 4604 elco 4753 elcnv 4764 dfmpt3 5293 0neqopab 5867 brabvv 5868 opabex3d 6070 opabex3 6071 |
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