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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 2566 | . 2 | |
2 | vex 2560 | . . . . . 6 | |
3 | vex 2560 | . . . . . 6 | |
4 | 2, 3 | opex 3966 | . . . . 5 |
5 | eleq1 2100 | . . . . 5 | |
6 | 4, 5 | mpbiri 157 | . . . 4 |
7 | 6 | adantr 261 | . . 3 |
8 | 7 | exlimivv 1776 | . 2 |
9 | eqeq1 2046 | . . . . 5 | |
10 | 9 | anbi1d 438 | . . . 4 |
11 | 10 | 2exbidv 1748 | . . 3 |
12 | df-opab 3819 | . . 3 | |
13 | 11, 12 | elab2g 2689 | . 2 |
14 | 1, 8, 13 | pm5.21nii 620 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 97 wb 98 wceq 1243 wex 1381 wcel 1393 cvv 2557 cop 3378 copab 3817 |
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 |
This theorem is referenced by: opelopabsbALT 3996 opelopabsb 3997 opelopabt 3999 opelopabga 4000 opabm 4017 iunopab 4018 epelg 4027 elxp 4362 elcnv 4512 dfmpt3 5021 0neqopab 5550 brabvv 5551 opabex3d 5748 opabex3 5749 |
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