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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 2610 | . 2 | |
2 | vex 2604 | . . . . . 6 | |
3 | vex 2604 | . . . . . 6 | |
4 | 2, 3 | opex 3984 | . . . . 5 |
5 | eleq1 2141 | . . . . 5 | |
6 | 4, 5 | mpbiri 166 | . . . 4 |
7 | 6 | adantr 270 | . . 3 |
8 | 7 | exlimivv 1817 | . 2 |
9 | eqeq1 2087 | . . . . 5 | |
10 | 9 | anbi1d 452 | . . . 4 |
11 | 10 | 2exbidv 1789 | . . 3 |
12 | df-opab 3840 | . . 3 | |
13 | 11, 12 | elab2g 2740 | . 2 |
14 | 1, 8, 13 | pm5.21nii 652 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 102 wb 103 wceq 1284 wex 1421 wcel 1433 cvv 2601 cop 3401 copab 3838 |
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 |
This theorem is referenced by: opelopabsbALT 4014 opelopabsb 4015 opelopabt 4017 opelopabga 4018 opabm 4035 iunopab 4036 epelg 4045 elxp 4380 elcnv 4530 dfmpt3 5041 0neqopab 5570 brabvv 5571 opabex3d 5768 opabex3 5769 |
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