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Mirrors > Home > ILE Home > Th. List > xpexg | Unicode version |
Description: The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. (Contributed by NM, 14-Aug-1994.) |
Ref | Expression |
---|---|
xpexg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpsspw 4710 | . 2 | |
2 | unexg 4415 | . . 3 | |
3 | pwexg 4153 | . . 3 | |
4 | pwexg 4153 | . . 3 | |
5 | 2, 3, 4 | 3syl 17 | . 2 |
6 | ssexg 4115 | . 2 | |
7 | 1, 5, 6 | sylancr 411 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 2135 cvv 2721 cun 3109 wss 3111 cpw 3553 cxp 4596 |
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-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-rex 2448 df-v 2723 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-opab 4038 df-xp 4604 |
This theorem is referenced by: xpex 4713 sqxpexg 4714 resiexg 4923 cnvexg 5135 coexg 5142 fex2 5350 fabexg 5369 resfunexgALT 6070 cofunexg 6071 fnexALT 6073 opabex3d 6081 opabex3 6082 oprabexd 6087 ofmresex 6097 mpoexxg 6170 tposexg 6217 erex 6516 pmex 6610 mapex 6611 pmvalg 6616 elpmg 6621 fvdiagfn 6650 ixpexgg 6679 ixpsnf1o 6693 map1 6769 xpdom2 6788 xpdom3m 6791 xpen 6802 mapxpen 6805 xpfi 6886 djuex 6999 djuassen 7164 cc2lem 7198 shftfvalg 10746 climconst2 11218 lmfval 12739 txbasex 12804 txopn 12812 txcn 12822 txrest 12823 blfvalps 12932 xmetxp 13054 limccnp2lem 13192 limccnp2cntop 13193 dvfvalap 13197 |
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