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Mirrors > Home > MPE Home > Th. List > peano2 | Structured version Visualization version 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 | peano2b 7081 | . 2 | |
2 | 1 | biimpi 206 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wcel 1990 csuc 5725 com 7065 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-tr 4753 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-om 7066 |
This theorem is referenced by: onnseq 7441 seqomlem1 7545 seqomlem4 7548 onasuc 7608 onmsuc 7609 onesuc 7610 nnacl 7691 nnecl 7693 nnacom 7697 nnmsucr 7705 1onn 7719 2onn 7720 3onn 7721 4onn 7722 nnneo 7731 nneob 7732 omopthlem1 7735 onomeneq 8150 dif1en 8193 findcard 8199 findcard2 8200 unbnn2 8217 dffi3 8337 wofib 8450 axinf2 8537 dfom3 8544 noinfep 8557 cantnflt 8569 trcl 8604 cardsucnn 8811 dif1card 8833 fseqdom 8849 alephfp 8931 ackbij1lem16 9057 ackbij2lem2 9062 ackbij2lem3 9063 ackbij2 9065 sornom 9099 infpssrlem4 9128 fin23lem26 9147 fin23lem20 9159 fin23lem38 9171 fin23lem39 9172 isf32lem2 9176 isf32lem3 9177 isf34lem7 9201 isf34lem6 9202 fin1a2lem6 9227 fin1a2lem9 9230 fin1a2lem12 9233 domtriomlem 9264 axdc2lem 9270 axdc3lem 9272 axdc3lem2 9273 axdc3lem4 9275 axdc4lem 9277 axdclem2 9342 peano2nn 11032 om2uzrani 12751 uzrdgsuci 12759 fzennn 12767 axdc4uzlem 12782 bnj970 31017 trpredtr 31730 elhf2 32282 0hf 32284 hfsn 32286 hfpw 32292 neibastop2lem 32355 finxpsuclem 33234 |
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