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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 3900 | . . 3 | |
2 | relxp 4618 | . . . . . 6 | |
3 | 2 | rgenw 2464 | . . . . 5 |
4 | reliun 4630 | . . . . 5 | |
5 | 3, 4 | mpbir 145 | . . . 4 |
6 | 5 | brrelex1i 4552 | . . 3 |
7 | 1, 6 | sylbir 134 | . 2 |
8 | elex 2671 | . . 3 | |
9 | 8 | adantr 274 | . 2 |
10 | nfcv 2258 | . . 3 | |
11 | nfiu1 3813 | . . . . 5 | |
12 | 11 | nfel2 2271 | . . . 4 |
13 | nfv 1493 | . . . 4 | |
14 | 12, 13 | nfbi 1553 | . . 3 |
15 | opeq1 3675 | . . . . 5 | |
16 | 15 | eleq1d 2186 | . . . 4 |
17 | eleq1 2180 | . . . . 5 | |
18 | opeliunxp2.1 | . . . . . 6 | |
19 | 18 | eleq2d 2187 | . . . . 5 |
20 | 17, 19 | anbi12d 464 | . . . 4 |
21 | 16, 20 | bibi12d 234 | . . 3 |
22 | opeliunxp 4564 | . . 3 | |
23 | 10, 14, 21, 22 | vtoclgf 2718 | . 2 |
24 | 7, 9, 23 | pm5.21nii 678 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1316 wcel 1465 wral 2393 cvv 2660 csn 3497 cop 3500 ciun 3783 class class class wbr 3899 cxp 4507 wrel 4514 |
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-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-csb 2976 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-iun 3785 df-br 3900 df-opab 3960 df-xp 4515 df-rel 4516 |
This theorem is referenced by: mpoxopn0yelv 6104 eldvap 12731 |
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