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Mirrors > Home > ILE Home > Th. List > xp1st | Unicode version |
Description: Location of the first element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
xp1st |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxp 4564 | . 2 | |
2 | vex 2692 | . . . . . . 7 | |
3 | vex 2692 | . . . . . . 7 | |
4 | 2, 3 | op1std 6054 | . . . . . 6 |
5 | 4 | eleq1d 2209 | . . . . 5 |
6 | 5 | biimpar 295 | . . . 4 |
7 | 6 | adantrr 471 | . . 3 |
8 | 7 | exlimivv 1869 | . 2 |
9 | 1, 8 | sylbi 120 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1332 wex 1469 wcel 1481 cop 3535 cxp 4545 cfv 5131 c1st 6044 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-iota 5096 df-fun 5133 df-fv 5139 df-1st 6046 |
This theorem is referenced by: disjxp1 6141 xpf1o 6746 xpmapenlem 6751 djuf1olem 6946 eldju1st 6964 dfplpq2 7186 dfmpq2 7187 enqbreq2 7189 enqdc1 7194 mulpipq2 7203 preqlu 7304 elnp1st2nd 7308 cauappcvgprlemladd 7490 elreal2 7662 cnref1o 9469 frecuzrdgrrn 10212 frec2uzrdg 10213 frecuzrdgrcl 10214 frecuzrdgsuc 10218 frecuzrdgrclt 10219 frecuzrdgg 10220 frecuzrdgsuctlem 10227 seq3val 10262 seqvalcd 10263 fsum2dlemstep 11235 fisumcom2 11239 eucalgval 11771 eucalginv 11773 eucalglt 11774 eucalg 11776 sqpweven 11889 2sqpwodd 11890 ctiunctlemudc 11986 tx2cn 12478 txdis 12485 txhmeo 12527 xmetxp 12715 xmetxpbl 12716 xmettxlem 12717 xmettx 12718 |
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