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Mirrors > Home > ILE Home > Th. List > peano2nn | Unicode version |
Description: Peano postulate: a successor of a positive integer is a positive integer. (Contributed by NM, 11-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.) |
Ref | Expression |
---|---|
peano2nn |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfnn2 8919 |
. . . . . 6
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2 | 1 | eleq2i 2244 |
. . . . 5
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3 | elintg 3852 |
. . . . 5
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4 | 2, 3 | bitrid 192 |
. . . 4
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5 | 4 | ibi 176 |
. . 3
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6 | vex 2740 |
. . . . . . . 8
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7 | eleq2 2241 |
. . . . . . . . 9
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8 | eleq2 2241 |
. . . . . . . . . 10
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9 | 8 | raleqbi1dv 2680 |
. . . . . . . . 9
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10 | 7, 9 | anbi12d 473 |
. . . . . . . 8
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11 | 6, 10 | elab 2881 |
. . . . . . 7
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12 | 11 | simprbi 275 |
. . . . . 6
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13 | oveq1 5881 |
. . . . . . . 8
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14 | 13 | eleq1d 2246 |
. . . . . . 7
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15 | 14 | rspcva 2839 |
. . . . . 6
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16 | 12, 15 | sylan2 286 |
. . . . 5
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17 | 16 | expcom 116 |
. . . 4
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18 | 17 | ralimia 2538 |
. . 3
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19 | 5, 18 | syl 14 |
. 2
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20 | nnre 8924 |
. . . 4
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21 | 1red 7971 |
. . . 4
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22 | 20, 21 | readdcld 7985 |
. . 3
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23 | 1 | eleq2i 2244 |
. . . 4
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24 | elintg 3852 |
. . . 4
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25 | 23, 24 | bitrid 192 |
. . 3
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26 | 22, 25 | syl 14 |
. 2
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27 | 19, 26 | mpbird 167 |
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-ext 2159 ax-sep 4121 ax-cnex 7901 ax-resscn 7902 ax-1re 7904 ax-addrcl 7907 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4004 df-iota 5178 df-fv 5224 df-ov 5877 df-inn 8918 |
This theorem is referenced by: peano2nnd 8932 nnind 8933 nnaddcl 8937 2nn 9078 3nn 9079 4nn 9080 5nn 9081 6nn 9082 7nn 9083 8nn 9084 9nn 9085 nneoor 9353 10nn 9397 nnsplit 10134 fzonn0p1p1 10210 expp1 10524 facp1 10705 resqrexlemfp1 11013 resqrexlemcalc3 11020 trireciplem 11503 trirecip 11504 cvgratnnlemnexp 11527 cvgratz 11535 nno 11905 nnoddm1d2 11909 rplpwr 12022 prmind2 12114 sqrt2irr 12156 pcmpt 12335 pockthi 12350 mulgnnp1 12945 2sqlem10 14354 |
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