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Mirrors > Home > MPE Home > Th. List > peano1 | Structured version Visualization version Unicode version |
Description: Zero is a natural number. One of Peano's five postulates for arithmetic. Proposition 7.30(1) of [TakeutiZaring] p. 42. Note: Unlike most textbooks, our proofs of peano1 7085 through peano5 7089 do not use the Axiom of Infinity. Unlike Takeuti and Zaring, they also do not use the Axiom of Regularity. (Contributed by NM, 15-May-1994.) |
Ref | Expression |
---|---|
peano1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limom 7080 | . 2 | |
2 | 0ellim 5787 | . 2 | |
3 | 1, 2 | ax-mp 5 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wcel 1990 c0 3915 wlim 5724 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 rdg0 7517 fr0g 7531 seqomlem3 7547 oa1suc 7611 om1 7622 oe1 7624 nna0r 7689 nnm0r 7690 nnmcl 7692 nnecl 7693 nnmsucr 7705 nnaword1 7709 nnaordex 7718 1onn 7719 oaabs2 7725 nnm1 7728 nneob 7732 omopth 7738 snfi 8038 0sdom1dom 8158 0fin 8188 findcard2 8200 nnunifi 8211 unblem2 8213 infn0 8222 unfilem3 8226 dffi3 8337 inf0 8518 infeq5i 8533 axinf2 8537 dfom3 8544 infdifsn 8554 noinfep 8557 cantnflt 8569 cnfcomlem 8596 cnfcom 8597 cnfcom2lem 8598 cnfcom3lem 8600 cnfcom3 8601 trcl 8604 rankdmr1 8664 rankeq0b 8723 cardlim 8798 infxpenc 8841 infxpenc2 8845 alephgeom 8905 alephfplem4 8930 ackbij1lem13 9054 ackbij1 9060 ackbij1b 9061 ominf4 9134 fin23lem16 9157 fin23lem31 9165 fin23lem40 9173 isf32lem9 9183 isf34lem7 9201 isf34lem6 9202 fin1a2lem6 9227 fin1a2lem7 9228 fin1a2lem11 9232 axdc3lem2 9273 axdc3lem4 9275 axdc4lem 9277 axcclem 9279 axdclem2 9342 pwfseqlem5 9485 omina 9513 wunex3 9563 1lt2pi 9727 1nn 11031 om2uzrani 12751 uzrdg0i 12758 fzennn 12767 axdc4uzlem 12782 hash1 13192 ltbwe 19472 2ndcdisj2 21260 snct 29491 trpredpred 31728 0hf 32284 neibastop2lem 32355 rdgeqoa 33218 finxp0 33228 |
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