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Mirrors > Home > MPE Home > Th. List > elvv | Structured version Visualization version 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 5542 | . 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))) | |
2 | vex 3444 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | vex 3444 | . . . . 5 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | pm3.2i 474 | . . . 4 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V) |
5 | 4 | biantru 533 | . . 3 ⊢ (𝐴 = 〈𝑥, 𝑦〉 ↔ (𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))) |
6 | 5 | 2exbii 1850 | . 2 ⊢ (∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉 ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))) |
7 | 1, 6 | bitr4i 281 | 1 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 Vcvv 3441 〈cop 4531 × cxp 5517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-opab 5093 df-xp 5525 |
This theorem is referenced by: elvvv 5591 elvvuni 5592 elopaelxp 5605 elrel 5635 copsex2gb 5643 relop 5685 elreldm 5769 dmsnn0 6031 funsndifnop 6890 1stval2 7688 2ndval2 7689 1st2val 7699 2nd2val 7700 dfopab2 7732 dfoprab3s 7733 dftpos4 7894 tpostpos 7895 fundmen 8566 fundmge2nop0 13846 ssrelf 30379 dfdm5 33129 dfrn5 33130 brtxp2 33455 pprodss4v 33458 brpprod3a 33460 brimg 33511 brxrn2 35787 fun2dmnopgexmpl 43840 |
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