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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 2749 |
. 2
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2 | simpl 109 |
. . . . . 6
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3 | eleq1 2240 |
. . . . . . . 8
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4 | suceq 4403 |
. . . . . . . . 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 2789 |
. . . . 5
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32 | 31 | ralrimiva 2550 |
. . . 4
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33 | ralim 2536 |
. . . . 5
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34 | elintg 3853 |
. . . . . 6
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35 | sucexg 4498 |
. . . . . . 7
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36 | elintg 3853 |
. . . . . . 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 4592 |
. . . 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 4122 ax-pow 4175 ax-pr 4210 ax-un 4434 |
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 2740 df-un 3134 df-in 3136 df-ss 3143 df-pw 3578 df-sn 3599 df-pr 3600 df-uni 3811 df-int 3846 df-suc 4372 df-iom 4591 |
This theorem is referenced by: peano5 4598 limom 4614 peano2b 4615 nnregexmid 4621 omsinds 4622 freccllem 6403 frecfcllem 6405 frecsuclem 6407 frecrdg 6409 nnacl 6481 nnacom 6485 nnmsucr 6489 nnsucsssuc 6493 nnaword 6512 1onn 6521 2onn 6522 3onn 6523 4onn 6524 nnaordex 6529 php5 6858 phplem4dom 6862 php5dom 6863 phplem4on 6867 dif1en 6879 findcard 6888 findcard2 6889 findcard2s 6890 infnfi 6895 unsnfi 6918 omp1eomlem 7093 ctmlemr 7107 infnninf 7122 infnninfOLD 7123 nnnninf 7124 nnnninfeq 7126 nninfwlpoimlemg 7173 nninfwlpoimlemginf 7174 frec2uzrand 10405 frecuzrdgsuc 10414 frecuzrdgsuctlem 10423 frecfzennn 10426 hashunlem 10784 ennnfonelemk 12401 ennnfonelemg 12404 ennnfonelemkh 12413 ennnfonelemhf1o 12414 ennnfonelemex 12415 ennnfonelemrn 12420 ennnfonelemnn0 12423 ctinfomlemom 12428 0nninf 14756 nnsf 14757 peano4nninf 14758 nninfsellemdc 14762 nninfsellemsuc 14764 nninfself 14765 nninfsellemeqinf 14768 |
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