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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2746 | . 2 | |
2 | simpl 109 | . . . . . 6 | |
3 | eleq1 2238 | . . . . . . . 8 | |
4 | suceq 4396 | . . . . . . . . 9 | |
5 | 4 | eleq1d 2244 | . . . . . . . 8 |
6 | 3, 5 | imbi12d 234 | . . . . . . 7 |
7 | 6 | adantl 277 | . . . . . 6 |
8 | df-clab 2162 | . . . . . . . . 9 | |
9 | simpr 110 | . . . . . . . . . . . 12 | |
10 | df-ral 2458 | . . . . . . . . . . . 12 | |
11 | 9, 10 | sylib 122 | . . . . . . . . . . 11 |
12 | 11 | sbimi 1762 | . . . . . . . . . 10 |
13 | sbim 1951 | . . . . . . . . . . . 12 | |
14 | clelsb2 2281 | . . . . . . . . . . . . 13 | |
15 | clelsb2 2281 | . . . . . . . . . . . . 13 | |
16 | 14, 15 | imbi12i 239 | . . . . . . . . . . . 12 |
17 | 13, 16 | bitri 184 | . . . . . . . . . . 11 |
18 | 17 | sbalv 2003 | . . . . . . . . . 10 |
19 | 12, 18 | sylib 122 | . . . . . . . . 9 |
20 | 8, 19 | sylbi 121 | . . . . . . . 8 |
21 | 20 | 19.21bi 1556 | . . . . . . 7 |
22 | 21 | adantl 277 | . . . . . 6 |
23 | nfv 1526 | . . . . . . 7 | |
24 | nfv 1526 | . . . . . . . . 9 | |
25 | nfra1 2506 | . . . . . . . . 9 | |
26 | 24, 25 | nfan 1563 | . . . . . . . 8 |
27 | 26 | nfsab 2167 | . . . . . . 7 |
28 | 23, 27 | nfan 1563 | . . . . . 6 |
29 | nfcvd 2318 | . . . . . 6 | |
30 | nfvd 1527 | . . . . . 6 | |
31 | 2, 7, 22, 28, 29, 30 | vtocldf 2786 | . . . . 5 |
32 | 31 | ralrimiva 2548 | . . . 4 |
33 | ralim 2534 | . . . . 5 | |
34 | elintg 3848 | . . . . . 6 | |
35 | sucexg 4491 | . . . . . . 7 | |
36 | elintg 3848 | . . . . . . 7 | |
37 | 35, 36 | syl 14 | . . . . . 6 |
38 | 34, 37 | imbi12d 234 | . . . . 5 |
39 | 33, 38 | syl5ibr 156 | . . . 4 |
40 | 32, 39 | mpd 13 | . . 3 |
41 | dfom3 4585 | . . . 4 | |
42 | 41 | eleq2i 2242 | . . 3 |
43 | 41 | eleq2i 2242 | . . 3 |
44 | 40, 42, 43 | 3imtr4g 205 | . 2 |
45 | 1, 44 | mpcom 36 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 104 wb 105 wal 1351 wceq 1353 wsb 1760 wcel 2146 cab 2161 wral 2453 cvv 2735 c0 3420 cint 3840 csuc 4359 com 4583 |
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 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-pow 4169 ax-pr 4203 ax-un 4427 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1459 df-sb 1761 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ral 2458 df-rex 2459 df-v 2737 df-un 3131 df-in 3133 df-ss 3140 df-pw 3574 df-sn 3595 df-pr 3596 df-uni 3806 df-int 3841 df-suc 4365 df-iom 4584 |
This theorem is referenced by: peano5 4591 limom 4607 peano2b 4608 nnregexmid 4614 omsinds 4615 freccllem 6393 frecfcllem 6395 frecsuclem 6397 frecrdg 6399 nnacl 6471 nnacom 6475 nnmsucr 6479 nnsucsssuc 6483 nnaword 6502 1onn 6511 2onn 6512 3onn 6513 4onn 6514 nnaordex 6519 php5 6848 phplem4dom 6852 php5dom 6853 phplem4on 6857 dif1en 6869 findcard 6878 findcard2 6879 findcard2s 6880 infnfi 6885 unsnfi 6908 omp1eomlem 7083 ctmlemr 7097 infnninf 7112 infnninfOLD 7113 nnnninf 7114 nnnninfeq 7116 nninfwlpoimlemg 7163 nninfwlpoimlemginf 7164 frec2uzrand 10373 frecuzrdgsuc 10382 frecuzrdgsuctlem 10391 frecfzennn 10394 hashunlem 10750 ennnfonelemk 12366 ennnfonelemg 12369 ennnfonelemkh 12378 ennnfonelemhf1o 12379 ennnfonelemex 12380 ennnfonelemrn 12385 ennnfonelemnn0 12388 ctinfomlemom 12393 0nninf 14312 nnsf 14313 peano4nninf 14314 nninfsellemdc 14318 nninfsellemsuc 14320 nninfself 14321 nninfsellemeqinf 14324 |
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