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Mirrors > Home > ILE Home > Th. List > elvv | GIF version |
Description: Membership in universal class of ordered pairs. (Contributed by NM, 4-Jul-1994.) |
Ref | Expression |
---|---|
elvv | ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxp 4615 | . 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))) | |
2 | vex 2724 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | vex 2724 | . . . . 5 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | pm3.2i 270 | . . . 4 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V) |
5 | 4 | biantru 300 | . . 3 ⊢ (𝐴 = 〈𝑥, 𝑦〉 ↔ (𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))) |
6 | 5 | 2exbii 1593 | . 2 ⊢ (∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉 ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))) |
7 | 1, 6 | bitr4i 186 | 1 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1342 ∃wex 1479 ∈ wcel 2135 Vcvv 2721 〈cop 3573 × cxp 4596 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-opab 4038 df-xp 4604 |
This theorem is referenced by: elvvv 4661 elvvuni 4662 ssrel 4686 elrel 4700 relop 4748 elreldm 4824 dmsnm 5063 1stval2 6115 2ndval2 6116 dfopab2 6149 dfoprab3s 6150 dftpos4 6222 tpostpos 6223 fundmen 6763 |
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