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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 4955 | . 2 | |
2 | dmxpm 4754 | . . . . . 6 | |
3 | 2 | adantl 275 | . . . . 5 |
4 | dmexg 4798 | . . . . . 6 | |
5 | 4 | adantr 274 | . . . . 5 |
6 | 3, 5 | eqeltrrd 2215 | . . . 4 |
7 | rnxpm 4963 | . . . . . 6 | |
8 | 7 | adantl 275 | . . . . 5 |
9 | rnexg 4799 | . . . . . 6 | |
10 | 9 | adantr 274 | . . . . 5 |
11 | 8, 10 | eqeltrrd 2215 | . . . 4 |
12 | 6, 11 | anim12dan 589 | . . 3 |
13 | 12 | ancom2s 555 | . 2 |
14 | 1, 13 | sylan2br 286 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wex 1468 wcel 1480 cvv 2681 cxp 4532 cdm 4534 crn 4535 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-xp 4540 df-rel 4541 df-cnv 4542 df-dm 4544 df-rn 4545 |
This theorem is referenced by: (None) |
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