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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 2771 |
. 2
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2 | simpl 109 |
. . . . . 6
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3 | eleq1 2256 |
. . . . . . . 8
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4 | suceq 4433 |
. . . . . . . . 9
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5 | 4 | eleq1d 2262 |
. . . . . . . 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 2180 |
. . . . . . . . 9
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9 | simpr 110 |
. . . . . . . . . . . 12
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10 | df-ral 2477 |
. . . . . . . . . . . 12
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11 | 9, 10 | sylib 122 |
. . . . . . . . . . 11
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12 | 11 | sbimi 1775 |
. . . . . . . . . 10
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13 | sbim 1969 |
. . . . . . . . . . . 12
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14 | clelsb2 2299 |
. . . . . . . . . . . . 13
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15 | clelsb2 2299 |
. . . . . . . . . . . . 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 2021 |
. . . . . . . . . 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 1569 |
. . . . . . 7
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22 | 21 | adantl 277 |
. . . . . 6
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23 | nfv 1539 |
. . . . . . 7
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24 | nfv 1539 |
. . . . . . . . 9
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25 | nfra1 2525 |
. . . . . . . . 9
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26 | 24, 25 | nfan 1576 |
. . . . . . . 8
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27 | 26 | nfsab 2185 |
. . . . . . 7
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28 | 23, 27 | nfan 1576 |
. . . . . 6
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29 | nfcvd 2337 |
. . . . . 6
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30 | nfvd 1540 |
. . . . . 6
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31 | 2, 7, 22, 28, 29, 30 | vtocldf 2811 |
. . . . 5
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32 | 31 | ralrimiva 2567 |
. . . 4
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33 | ralim 2553 |
. . . . 5
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34 | elintg 3878 |
. . . . . 6
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35 | sucexg 4530 |
. . . . . . 7
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36 | elintg 3878 |
. . . . . . 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 4624 |
. . . 4
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42 | 41 | eleq2i 2260 |
. . 3
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43 | 41 | eleq2i 2260 |
. . 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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4464 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-v 2762 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-uni 3836 df-int 3871 df-suc 4402 df-iom 4623 |
This theorem is referenced by: peano5 4630 limom 4646 peano2b 4647 nnregexmid 4653 omsinds 4654 freccllem 6455 frecfcllem 6457 frecsuclem 6459 frecrdg 6461 nnacl 6533 nnacom 6537 nnmsucr 6541 nnsucsssuc 6545 nnaword 6564 1onn 6573 2onn 6574 3onn 6575 4onn 6576 nnaordex 6581 php5 6914 phplem4dom 6918 php5dom 6919 phplem4on 6923 dif1en 6935 findcard 6944 findcard2 6945 findcard2s 6946 infnfi 6951 unsnfi 6975 omp1eomlem 7153 ctmlemr 7167 nninfninc 7182 infnninf 7183 infnninfOLD 7184 nnnninf 7185 nnnninfeq 7187 nninfwlpoimlemg 7234 nninfwlpoimlemginf 7235 frec2uzrand 10476 frecuzrdgsuc 10485 frecuzrdgsuctlem 10494 frecfzennn 10497 hashunlem 10875 ennnfonelemk 12557 ennnfonelemg 12560 ennnfonelemkh 12569 ennnfonelemhf1o 12570 ennnfonelemex 12571 ennnfonelemrn 12576 ennnfonelemnn0 12579 ctinfomlemom 12584 0nninf 15494 nnsf 15495 peano4nninf 15496 nninfsellemdc 15500 nninfsellemsuc 15502 nninfself 15503 nninfsellemeqinf 15506 |
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