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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 6130 | . 2 | |
2 | elvvuni 4662 | . . . 4 | |
3 | 2 | adantr 274 | . . 3 |
4 | simprl 521 | . . . . . 6 | |
5 | eleq2 2228 | . . . . . . . 8 | |
6 | eleq1 2227 | . . . . . . . . 9 | |
7 | fveq2 5480 | . . . . . . . . . . 11 | |
8 | 7 | eleq1d 2233 | . . . . . . . . . 10 |
9 | fveq2 5480 | . . . . . . . . . . 11 | |
10 | 9 | eleq1d 2233 | . . . . . . . . . 10 |
11 | 8, 10 | anbi12d 465 | . . . . . . . . 9 |
12 | 6, 11 | anbi12d 465 | . . . . . . . 8 |
13 | 5, 12 | anbi12d 465 | . . . . . . 7 |
14 | 13 | spcegv 2809 | . . . . . 6 |
15 | 4, 14 | mpcom 36 | . . . . 5 |
16 | eluniab 3795 | . . . . 5 | |
17 | 15, 16 | sylibr 133 | . . . 4 |
18 | xp2 6133 | . . . . . 6 | |
19 | df-rab 2451 | . . . . . 6 | |
20 | 18, 19 | eqtri 2185 | . . . . 5 |
21 | 20 | unieqi 3793 | . . . 4 |
22 | 17, 21 | eleqtrrdi 2258 | . . 3 |
23 | 3, 22 | mpancom 419 | . 2 |
24 | 1, 23 | sylbi 120 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1342 wex 1479 wcel 2135 cab 2150 crab 2446 cvv 2721 cuni 3783 cxp 4596 cfv 5182 c1st 6098 c2nd 6099 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fo 5188 df-fv 5190 df-1st 6100 df-2nd 6101 |
This theorem is referenced by: (None) |
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