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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 4484 |
. 2
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2 | vex 2636 |
. . . . . . 7
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3 | vex 2636 |
. . . . . . 7
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4 | 2, 3 | op1std 5957 |
. . . . . 6
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5 | 4 | eleq1d 2163 |
. . . . 5
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6 | 5 | biimpar 292 |
. . . 4
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7 | 6 | adantrr 464 |
. . 3
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8 | 7 | exlimivv 1831 |
. 2
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9 | 1, 8 | sylbi 120 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-v 2635 df-sbc 2855 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-opab 3922 df-mpt 3923 df-id 4144 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-iota 5014 df-fun 5051 df-fv 5057 df-1st 5949 |
This theorem is referenced by: disjxp1 6039 xpf1o 6640 xpmapenlem 6645 djuf1olem 6825 djur 6837 eldju1st 6842 dfplpq2 7010 dfmpq2 7011 enqbreq2 7013 enqdc1 7018 mulpipq2 7027 preqlu 7128 elnp1st2nd 7132 cauappcvgprlemladd 7314 elreal2 7465 cnref1o 9232 frecuzrdgrrn 9964 frec2uzrdg 9965 frecuzrdgrcl 9966 frecuzrdgsuc 9970 frecuzrdgrclt 9971 frecuzrdgg 9972 frecuzrdgsuctlem 9979 iseqvalt 10019 seq3val 10020 fsum2dlemstep 10992 fisumcom2 10996 eucalgval 11478 eucalginv 11480 eucalglt 11481 eucalg 11483 sqpweven 11595 2sqpwodd 11596 |
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