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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxp 4645 |
. 2
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2 | vex 2742 |
. . . . . . 7
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3 | vex 2742 |
. . . . . . 7
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4 | 2, 3 | op1std 6151 |
. . . . . 6
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5 | 4 | eleq1d 2246 |
. . . . 5
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6 | 5 | biimpar 297 |
. . . 4
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7 | 6 | adantrr 479 |
. . 3
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8 | 7 | exlimivv 1896 |
. 2
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9 | 1, 8 | sylbi 121 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2741 df-sbc 2965 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-iota 5180 df-fun 5220 df-fv 5226 df-1st 6143 |
This theorem is referenced by: disjxp1 6239 xpf1o 6846 xpmapenlem 6851 djuf1olem 7054 eldju1st 7072 exmidapne 7261 dfplpq2 7355 dfmpq2 7356 enqbreq2 7358 enqdc1 7363 mulpipq2 7372 preqlu 7473 elnp1st2nd 7477 cauappcvgprlemladd 7659 elreal2 7831 cnref1o 9652 frecuzrdgrrn 10410 frec2uzrdg 10411 frecuzrdgrcl 10412 frecuzrdgsuc 10416 frecuzrdgrclt 10417 frecuzrdgg 10418 frecuzrdgsuctlem 10425 seq3val 10460 seqvalcd 10461 fsum2dlemstep 11444 fisumcom2 11448 fprod2dlemstep 11632 fprodcom2fi 11636 eucalgval 12056 eucalginv 12058 eucalglt 12059 eucalg 12061 sqpweven 12177 2sqpwodd 12178 ctiunctlemudc 12440 xpsff1o 12773 tx2cn 13855 txdis 13862 txhmeo 13904 xmetxp 14092 xmetxpbl 14093 xmettxlem 14094 xmettx 14095 |
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