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Mirrors > Home > ILE Home > Th. List > xpexr2m | Unicode version |
Description: If a nonempty cross product is a set, so are both of its components. (Contributed by Jim Kingdon, 14-Dec-2018.) |
Ref | Expression |
---|---|
xpexr2m |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpm 4765 | . 2 | |
2 | dmxpm 4573 | . . . . . 6 | |
3 | 2 | adantl 271 | . . . . 5 |
4 | dmexg 4614 | . . . . . 6 | |
5 | 4 | adantr 270 | . . . . 5 |
6 | 3, 5 | eqeltrrd 2156 | . . . 4 |
7 | rnxpm 4772 | . . . . . 6 | |
8 | 7 | adantl 271 | . . . . 5 |
9 | rnexg 4615 | . . . . . 6 | |
10 | 9 | adantr 270 | . . . . 5 |
11 | 8, 10 | eqeltrrd 2156 | . . . 4 |
12 | 6, 11 | anim12dan 564 | . . 3 |
13 | 12 | ancom2s 530 | . 2 |
14 | 1, 13 | sylan2br 282 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wceq 1284 wex 1421 wcel 1433 cvv 2601 cxp 4361 cdm 4363 crn 4364 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-xp 4369 df-rel 4370 df-cnv 4371 df-dm 4373 df-rn 4374 |
This theorem is referenced by: (None) |
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