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Mirrors > Home > ILE Home > Th. List > elopab | Unicode version |
Description: Membership in a class abstraction of pairs. (Contributed by NM, 24-Mar-1998.) |
Ref | Expression |
---|---|
elopab |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2671 | . 2 | |
2 | vex 2663 | . . . . . 6 | |
3 | vex 2663 | . . . . . 6 | |
4 | 2, 3 | opex 4121 | . . . . 5 |
5 | eleq1 2180 | . . . . 5 | |
6 | 4, 5 | mpbiri 167 | . . . 4 |
7 | 6 | adantr 274 | . . 3 |
8 | 7 | exlimivv 1852 | . 2 |
9 | eqeq1 2124 | . . . . 5 | |
10 | 9 | anbi1d 460 | . . . 4 |
11 | 10 | 2exbidv 1824 | . . 3 |
12 | df-opab 3960 | . . 3 | |
13 | 11, 12 | elab2g 2804 | . 2 |
14 | 1, 8, 13 | pm5.21nii 678 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1316 wex 1453 wcel 1465 cvv 2660 cop 3500 copab 3958 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-opab 3960 |
This theorem is referenced by: opelopabsbALT 4151 opelopabsb 4152 opelopabt 4154 opelopabga 4155 opabm 4172 iunopab 4173 epelg 4182 elxp 4526 elco 4675 elcnv 4686 dfmpt3 5215 0neqopab 5784 brabvv 5785 opabex3d 5987 opabex3 5988 |
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