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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 5708 | . 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))) | |
| 2 | vex 3484 | . . . . 5 ⊢ 𝑥 ∈ V | |
| 3 | vex 3484 | . . . . 5 ⊢ 𝑦 ∈ V | |
| 4 | 2, 3 | pm3.2i 470 | . . . 4 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V) |
| 5 | 4 | biantru 529 | . . 3 ⊢ (𝐴 = 〈𝑥, 𝑦〉 ↔ (𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))) |
| 6 | 5 | 2exbii 1849 | . 2 ⊢ (∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉 ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))) |
| 7 | 1, 6 | bitr4i 278 | 1 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 Vcvv 3480 〈cop 4632 × cxp 5683 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-opab 5206 df-xp 5691 |
| This theorem is referenced by: elvvv 5761 elvvuni 5762 elopaelxpOLD 5776 elrel 5808 copsex2gb 5816 relop 5861 elreldm 5946 dmsnn0 6227 funsndifnop 7171 1stval2 8031 2ndval2 8032 1st2val 8042 2nd2val 8043 dfopab2 8077 dfoprab3s 8078 dftpos4 8270 tpostpos 8271 fundmen 9071 cnvfi 9216 fundmge2nop0 14541 ssrelf 32627 fineqvac 35111 dfdm5 35773 dfrn5 35774 brtxp2 35882 pprodss4v 35885 brpprod3a 35887 brimg 35938 brxrn2 38376 fun2dmnopgexmpl 47296 |
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