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Mirrors > Home > ILE Home > Th. List > eliunxp | Unicode version |
Description: Membership in a union of cross products. Analogue of elxp 4362 for nonconstant . (Contributed by Mario Carneiro, 29-Dec-2014.) |
Ref | Expression |
---|---|
eliunxp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relxp 4447 | . . . . . 6 | |
2 | 1 | rgenw 2376 | . . . . 5 |
3 | reliun 4458 | . . . . 5 | |
4 | 2, 3 | mpbir 134 | . . . 4 |
5 | elrel 4442 | . . . 4 | |
6 | 4, 5 | mpan 400 | . . 3 |
7 | 6 | pm4.71ri 372 | . 2 |
8 | nfiu1 3687 | . . . 4 | |
9 | 8 | nfel2 2190 | . . 3 |
10 | 9 | 19.41 1576 | . 2 |
11 | 19.41v 1782 | . . . 4 | |
12 | eleq1 2100 | . . . . . . 7 | |
13 | opeliunxp 4395 | . . . . . . 7 | |
14 | 12, 13 | syl6bb 185 | . . . . . 6 |
15 | 14 | pm5.32i 427 | . . . . 5 |
16 | 15 | exbii 1496 | . . . 4 |
17 | 11, 16 | bitr3i 175 | . . 3 |
18 | 17 | exbii 1496 | . 2 |
19 | 7, 10, 18 | 3bitr2i 197 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 97 wb 98 wceq 1243 wex 1381 wcel 1393 wral 2306 csn 3375 cop 3378 ciun 3657 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-opab 3819 df-xp 4351 df-rel 4352 |
This theorem is referenced by: raliunxp 4477 rexiunxp 4478 dfmpt3 5021 mpt2mptx 5595 |
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