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Mirrors > Home > ILE Home > Th. List > unielxp | Unicode version |
Description: The membership relation for a cross product is inherited by union. (Contributed by NM, 16-Sep-2006.) |
Ref | Expression |
---|---|
unielxp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxp7 5797 | . 2 | |
2 | elvvuni 4404 | . . . 4 | |
3 | 2 | adantr 261 | . . 3 |
4 | simprl 483 | . . . . . 6 | |
5 | eleq2 2101 | . . . . . . . 8 | |
6 | eleq1 2100 | . . . . . . . . 9 | |
7 | fveq2 5178 | . . . . . . . . . . 11 | |
8 | 7 | eleq1d 2106 | . . . . . . . . . 10 |
9 | fveq2 5178 | . . . . . . . . . . 11 | |
10 | 9 | eleq1d 2106 | . . . . . . . . . 10 |
11 | 8, 10 | anbi12d 442 | . . . . . . . . 9 |
12 | 6, 11 | anbi12d 442 | . . . . . . . 8 |
13 | 5, 12 | anbi12d 442 | . . . . . . 7 |
14 | 13 | spcegv 2641 | . . . . . 6 |
15 | 4, 14 | mpcom 32 | . . . . 5 |
16 | eluniab 3592 | . . . . 5 | |
17 | 15, 16 | sylibr 137 | . . . 4 |
18 | xp2 5799 | . . . . . 6 | |
19 | df-rab 2315 | . . . . . 6 | |
20 | 18, 19 | eqtri 2060 | . . . . 5 |
21 | 20 | unieqi 3590 | . . . 4 |
22 | 17, 21 | syl6eleqr 2131 | . . 3 |
23 | 3, 22 | mpancom 399 | . 2 |
24 | 1, 23 | sylbi 114 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wceq 1243 wex 1381 wcel 1393 cab 2026 crab 2310 cvv 2557 cuni 3580 cxp 4343 cfv 4902 c1st 5765 c2nd 5766 |
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-13 1404 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 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-fo 4908 df-fv 4910 df-1st 5767 df-2nd 5768 |
This theorem is referenced by: (None) |
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