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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 4380 | . 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))) | |
2 | vex 2604 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | vex 2604 | . . . . 5 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | pm3.2i 266 | . . . 4 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V) |
5 | 4 | biantru 296 | . . 3 ⊢ (𝐴 = 〈𝑥, 𝑦〉 ↔ (𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))) |
6 | 5 | 2exbii 1537 | . 2 ⊢ (∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉 ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))) |
7 | 1, 6 | bitr4i 185 | 1 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 ↔ wb 103 = wceq 1284 ∃wex 1421 ∈ wcel 1433 Vcvv 2601 〈cop 3401 × cxp 4361 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-opab 3840 df-xp 4369 |
This theorem is referenced by: elvvv 4421 elvvuni 4422 ssrel 4446 elrel 4460 relop 4504 elreldm 4578 dmsnm 4806 1stval2 5802 2ndval2 5803 dfopab2 5835 dfoprab3s 5836 dftpos4 5901 tpostpos 5902 fundmen 6309 |
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