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Mirrors > Home > ILE Home > Th. List > xp2nd | Unicode version |
Description: Location of the second element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
xp2nd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxp 4604 | . 2 | |
2 | vex 2715 | . . . . . . 7 | |
3 | vex 2715 | . . . . . . 7 | |
4 | 2, 3 | op2ndd 6098 | . . . . . 6 |
5 | 4 | eleq1d 2226 | . . . . 5 |
6 | 5 | biimpar 295 | . . . 4 |
7 | 6 | adantrl 470 | . . 3 |
8 | 7 | exlimivv 1876 | . 2 |
9 | 1, 8 | sylbi 120 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1335 wex 1472 wcel 2128 cop 3563 cxp 4585 cfv 5171 c2nd 6088 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-pow 4136 ax-pr 4170 ax-un 4394 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-sbc 2938 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-br 3967 df-opab 4027 df-mpt 4028 df-id 4254 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-iota 5136 df-fun 5173 df-fv 5179 df-2nd 6090 |
This theorem is referenced by: xpf1o 6790 xpmapenlem 6795 djuf1olem 6998 cc2lem 7187 dfplpq2 7275 dfmpq2 7276 enqbreq2 7278 enqdc1 7283 mulpipq2 7292 preqlu 7393 elnp1st2nd 7397 cauappcvgprlemladd 7579 elreal2 7751 cnref1o 9560 frecuzrdgrrn 10311 frec2uzrdg 10312 frecuzrdgrcl 10313 frecuzrdgtcl 10315 frecuzrdgsuc 10317 frecuzrdgrclt 10318 frecuzrdgg 10319 frecuzrdgdomlem 10320 frecuzrdgfunlem 10322 frecuzrdgsuctlem 10326 seq3val 10361 seqvalcd 10362 fisumcom2 11339 fprodcom2fi 11527 eucalgval 11935 eucalginv 11937 eucalglt 11938 eucalgcvga 11939 eucalg 11940 sqpweven 12054 2sqpwodd 12055 ctiunctlemudc 12208 tx1cn 12711 txdis 12719 txhmeo 12761 xmetxp 12949 xmetxpbl 12950 xmettxlem 12951 xmettx 12952 |
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