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Mirrors > Home > ILE Home > Th. List > peano2 | Unicode version |
Description: The successor of any natural number is a natural number. One of Peano's five postulates for arithmetic. Proposition 7.30(2) of [TakeutiZaring] p. 42. (Contributed by NM, 3-Sep-2003.) |
Ref | Expression |
---|---|
peano2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2748 |
. 2
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2 | simpl 109 |
. . . . . 6
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3 | eleq1 2240 |
. . . . . . . 8
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4 | suceq 4402 |
. . . . . . . . 9
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5 | 4 | eleq1d 2246 |
. . . . . . . 8
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6 | 3, 5 | imbi12d 234 |
. . . . . . 7
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7 | 6 | adantl 277 |
. . . . . 6
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8 | df-clab 2164 |
. . . . . . . . 9
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9 | simpr 110 |
. . . . . . . . . . . 12
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10 | df-ral 2460 |
. . . . . . . . . . . 12
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11 | 9, 10 | sylib 122 |
. . . . . . . . . . 11
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12 | 11 | sbimi 1764 |
. . . . . . . . . 10
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13 | sbim 1953 |
. . . . . . . . . . . 12
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14 | clelsb2 2283 |
. . . . . . . . . . . . 13
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15 | clelsb2 2283 |
. . . . . . . . . . . . 13
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16 | 14, 15 | imbi12i 239 |
. . . . . . . . . . . 12
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17 | 13, 16 | bitri 184 |
. . . . . . . . . . 11
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18 | 17 | sbalv 2005 |
. . . . . . . . . 10
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19 | 12, 18 | sylib 122 |
. . . . . . . . 9
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20 | 8, 19 | sylbi 121 |
. . . . . . . 8
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21 | 20 | 19.21bi 1558 |
. . . . . . 7
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22 | 21 | adantl 277 |
. . . . . 6
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23 | nfv 1528 |
. . . . . . 7
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24 | nfv 1528 |
. . . . . . . . 9
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25 | nfra1 2508 |
. . . . . . . . 9
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26 | 24, 25 | nfan 1565 |
. . . . . . . 8
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27 | 26 | nfsab 2169 |
. . . . . . 7
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28 | 23, 27 | nfan 1565 |
. . . . . 6
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29 | nfcvd 2320 |
. . . . . 6
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30 | nfvd 1529 |
. . . . . 6
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31 | 2, 7, 22, 28, 29, 30 | vtocldf 2788 |
. . . . 5
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32 | 31 | ralrimiva 2550 |
. . . 4
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33 | ralim 2536 |
. . . . 5
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34 | elintg 3852 |
. . . . . 6
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35 | sucexg 4497 |
. . . . . . 7
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36 | elintg 3852 |
. . . . . . 7
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37 | 35, 36 | syl 14 |
. . . . . 6
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38 | 34, 37 | imbi12d 234 |
. . . . 5
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39 | 33, 38 | imbitrrid 156 |
. . . 4
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40 | 32, 39 | mpd 13 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
41 | dfom3 4591 |
. . . 4
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42 | 41 | eleq2i 2244 |
. . 3
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43 | 41 | eleq2i 2244 |
. . 3
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44 | 40, 42, 43 | 3imtr4g 205 |
. 2
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45 | 1, 44 | mpcom 36 |
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 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 |
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-pw 3577 df-sn 3598 df-pr 3599 df-uni 3810 df-int 3845 df-suc 4371 df-iom 4590 |
This theorem is referenced by: peano5 4597 limom 4613 peano2b 4614 nnregexmid 4620 omsinds 4621 freccllem 6402 frecfcllem 6404 frecsuclem 6406 frecrdg 6408 nnacl 6480 nnacom 6484 nnmsucr 6488 nnsucsssuc 6492 nnaword 6511 1onn 6520 2onn 6521 3onn 6522 4onn 6523 nnaordex 6528 php5 6857 phplem4dom 6861 php5dom 6862 phplem4on 6866 dif1en 6878 findcard 6887 findcard2 6888 findcard2s 6889 infnfi 6894 unsnfi 6917 omp1eomlem 7092 ctmlemr 7106 infnninf 7121 infnninfOLD 7122 nnnninf 7123 nnnninfeq 7125 nninfwlpoimlemg 7172 nninfwlpoimlemginf 7173 frec2uzrand 10404 frecuzrdgsuc 10413 frecuzrdgsuctlem 10422 frecfzennn 10425 hashunlem 10783 ennnfonelemk 12400 ennnfonelemg 12403 ennnfonelemkh 12412 ennnfonelemhf1o 12413 ennnfonelemex 12414 ennnfonelemrn 12419 ennnfonelemnn0 12422 ctinfomlemom 12427 0nninf 14723 nnsf 14724 peano4nninf 14725 nninfsellemdc 14729 nninfsellemsuc 14731 nninfself 14732 nninfsellemeqinf 14735 |
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