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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 4362 | . 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))) | |
2 | vex 2560 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | vex 2560 | . . . . 5 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | pm3.2i 257 | . . . 4 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V) |
5 | 4 | biantru 286 | . . 3 ⊢ (𝐴 = 〈𝑥, 𝑦〉 ↔ (𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))) |
6 | 5 | 2exbii 1497 | . 2 ⊢ (∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉 ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))) |
7 | 1, 6 | bitr4i 176 | 1 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 = wceq 1243 ∃wex 1381 ∈ wcel 1393 Vcvv 2557 〈cop 3378 × cxp 4343 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-opab 3819 df-xp 4351 |
This theorem is referenced by: elvvv 4403 elvvuni 4404 ssrel 4428 elrel 4442 relop 4486 elreldm 4560 dmsnm 4786 1stval2 5782 2ndval2 5783 dfopab2 5815 dfoprab3s 5816 dftpos4 5878 tpostpos 5879 fundmen 6286 |
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