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Mirrors > Home > ILE Home > Th. List > opeliunxp2f | Unicode version |
Description: Membership in a union of Cartesian products, using bound-variable hypothesis for instead of distinct variable conditions as in opeliunxp2 4749. (Contributed by AV, 25-Oct-2020.) |
Ref | Expression |
---|---|
opeliunxp2f.f | |
opeliunxp2f.e |
Ref | Expression |
---|---|
opeliunxp2f |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 3988 | . . 3 | |
2 | relxp 4718 | . . . . . 6 | |
3 | 2 | rgenw 2525 | . . . . 5 |
4 | reliun 4730 | . . . . 5 | |
5 | 3, 4 | mpbir 145 | . . . 4 |
6 | 5 | brrelex1i 4652 | . . 3 |
7 | 1, 6 | sylbir 134 | . 2 |
8 | elex 2741 | . . 3 | |
9 | 8 | adantr 274 | . 2 |
10 | nfiu1 3901 | . . . . 5 | |
11 | 10 | nfel2 2325 | . . . 4 |
12 | nfv 1521 | . . . . 5 | |
13 | opeliunxp2f.f | . . . . . 6 | |
14 | 13 | nfel2 2325 | . . . . 5 |
15 | 12, 14 | nfan 1558 | . . . 4 |
16 | 11, 15 | nfbi 1582 | . . 3 |
17 | opeq1 3763 | . . . . 5 | |
18 | 17 | eleq1d 2239 | . . . 4 |
19 | eleq1 2233 | . . . . 5 | |
20 | opeliunxp2f.e | . . . . . 6 | |
21 | 20 | eleq2d 2240 | . . . . 5 |
22 | 19, 21 | anbi12d 470 | . . . 4 |
23 | 18, 22 | bibi12d 234 | . . 3 |
24 | opeliunxp 4664 | . . 3 | |
25 | 16, 23, 24 | vtoclg1f 2789 | . 2 |
26 | 7, 9, 25 | pm5.21nii 699 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1348 wcel 2141 wnfc 2299 wral 2448 cvv 2730 csn 3581 cop 3584 ciun 3871 class class class wbr 3987 cxp 4607 wrel 4614 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-csb 3050 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-iun 3873 df-br 3988 df-opab 4049 df-xp 4615 df-rel 4616 |
This theorem is referenced by: fisumcom2 11394 fprodcom2fi 11582 |
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