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Mirrors > Home > ILE Home > Th. List > opeliunxp2 | Unicode version |
Description: Membership in a union of cross products. (Contributed by Mario Carneiro, 14-Feb-2015.) |
Ref | Expression |
---|---|
opeliunxp2.1 |
Ref | Expression |
---|---|
opeliunxp2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 3765 | . . 3 | |
2 | relxp 4447 | . . . . . 6 | |
3 | 2 | rgenw 2376 | . . . . 5 |
4 | reliun 4458 | . . . . 5 | |
5 | 3, 4 | mpbir 134 | . . . 4 |
6 | 5 | brrelexi 4384 | . . 3 |
7 | 1, 6 | sylbir 125 | . 2 |
8 | elex 2566 | . . 3 | |
9 | 8 | adantr 261 | . 2 |
10 | nfcv 2178 | . . 3 | |
11 | nfiu1 3687 | . . . . 5 | |
12 | 11 | nfel2 2190 | . . . 4 |
13 | nfv 1421 | . . . 4 | |
14 | 12, 13 | nfbi 1481 | . . 3 |
15 | opeq1 3549 | . . . . 5 | |
16 | 15 | eleq1d 2106 | . . . 4 |
17 | eleq1 2100 | . . . . 5 | |
18 | opeliunxp2.1 | . . . . . 6 | |
19 | 18 | eleq2d 2107 | . . . . 5 |
20 | 17, 19 | anbi12d 442 | . . . 4 |
21 | 16, 20 | bibi12d 224 | . . 3 |
22 | opeliunxp 4395 | . . 3 | |
23 | 10, 14, 21, 22 | vtoclgf 2612 | . 2 |
24 | 7, 9, 23 | pm5.21nii 620 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wcel 1393 wral 2306 cvv 2557 csn 3375 cop 3378 ciun 3657 class class class wbr 3764 cxp 4343 wrel 4350 |
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-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-csb 2853 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-iun 3659 df-br 3765 df-opab 3819 df-xp 4351 df-rel 4352 |
This theorem is referenced by: mpt2xopn0yelv 5854 |
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