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Mirrors > Home > ILE Home > Th. List > eluzp1p1 | Unicode version |
Description: Membership in the next upper set of integers. (Contributed by NM, 5-Oct-2005.) |
Ref | Expression |
---|---|
eluzp1p1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano2z 9285 |
. . . 4
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2 | 1 | 3ad2ant1 1018 |
. . 3
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3 | peano2z 9285 |
. . . 4
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4 | 3 | 3ad2ant2 1019 |
. . 3
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5 | zre 9253 |
. . . . 5
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6 | zre 9253 |
. . . . 5
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7 | 1re 7953 |
. . . . . 6
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8 | leadd1 8383 |
. . . . . 6
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9 | 7, 8 | mp3an3 1326 |
. . . . 5
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10 | 5, 6, 9 | syl2an 289 |
. . . 4
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11 | 10 | biimp3a 1345 |
. . 3
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12 | 2, 4, 11 | 3jca 1177 |
. 2
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13 | eluz2 9530 |
. 2
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14 | eluz2 9530 |
. 2
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15 | 12, 13, 14 | 3imtr4i 201 |
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-in1 614 ax-in2 615 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 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-cnex 7899 ax-resscn 7900 ax-1cn 7901 ax-1re 7902 ax-icn 7903 ax-addcl 7904 ax-addrcl 7905 ax-mulcl 7906 ax-addcom 7908 ax-addass 7910 ax-distr 7912 ax-i2m1 7913 ax-0id 7916 ax-rnegex 7917 ax-cnre 7919 ax-pre-ltadd 7924 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 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-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4003 df-opab 4064 df-mpt 4065 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5177 df-fun 5217 df-fn 5218 df-f 5219 df-fv 5223 df-riota 5828 df-ov 5875 df-oprab 5876 df-mpo 5877 df-pnf 7990 df-mnf 7991 df-xr 7992 df-ltxr 7993 df-le 7994 df-sub 8126 df-neg 8127 df-inn 8916 df-n0 9173 df-z 9250 df-uz 9525 |
This theorem is referenced by: uzp1 9557 fzp1elp1 10070 rebtwn2z 10250 seqvalcd 10454 seqovcd 10458 seqp1cd 10461 seq3fveq2 10464 seq3id2 10504 seq3coll 10815 serf0 11353 efcllemp 11659 prmind2 12112 pockthlem 12346 pockthg 12347 prmunb 12352 cvgcmp2nlemabs 14640 |
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