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Mirrors > Home > MPE Home > Th. List > peano1 | Structured version Visualization version GIF 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 7590 through peano5 7594 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 | ⊢ ∅ ∈ ω |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limom 7584 | . 2 ⊢ Lim ω | |
2 | 0ellim 6246 | . 2 ⊢ (Lim ω → ∅ ∈ ω) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ∅ ∈ ω |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 ∅c0 4288 Lim wlim 6185 ωcom 7569 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-tr 5164 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-om 7570 |
This theorem is referenced by: onnseq 7970 rdg0 8046 fr0g 8060 seqomlem3 8077 oa1suc 8145 o2p2e4 8155 om1 8157 oe1 8159 nna0r 8224 nnm0r 8225 nnmcl 8227 nnecl 8228 nnmsucr 8240 nnaword1 8244 nnaordex 8253 1onn 8254 oaabs2 8261 nnm1 8264 nneob 8268 omopth 8274 snfi 8582 0sdom1dom 8704 0fin 8734 findcard2 8746 nnunifi 8757 unblem2 8759 infn0 8768 unfilem3 8772 dffi3 8883 inf0 9072 infeq5i 9087 axinf2 9091 dfom3 9098 infdifsn 9108 noinfep 9111 cantnflt 9123 cnfcomlem 9150 cnfcom 9151 cnfcom2lem 9152 cnfcom3lem 9154 cnfcom3 9155 trcl 9158 rankdmr1 9218 rankeq0b 9277 cardlim 9389 infxpenc 9432 infxpenc2 9436 alephgeom 9496 alephfplem4 9521 ackbij1lem13 9642 ackbij1 9648 ackbij1b 9649 ominf4 9722 fin23lem16 9745 fin23lem31 9753 fin23lem40 9761 isf32lem9 9771 isf34lem7 9789 isf34lem6 9790 fin1a2lem6 9815 fin1a2lem7 9816 fin1a2lem11 9820 axdc3lem2 9861 axdc3lem4 9863 axdc4lem 9865 axcclem 9867 axdclem2 9930 pwfseqlem5 10073 omina 10101 wunex3 10151 1lt2pi 10315 1nn 11637 om2uzrani 13308 uzrdg0i 13315 fzennn 13324 axdc4uzlem 13339 hash1 13753 fnpr2o 16818 fvpr0o 16820 ltbwe 20181 2ndcdisj2 21993 snct 30375 goelel3xp 32492 satfv0 32502 satfv1 32507 satf0 32516 satf00 32518 satf0suclem 32519 sat1el2xp 32523 fmla0 32526 fmlasuc0 32528 fmla1 32531 gonan0 32536 gonar 32539 goalr 32541 satffunlem1lem2 32547 satffunlem1 32551 satefvfmla0 32562 prv0 32574 trpredpred 32964 0hf 33535 neibastop2lem 33605 rdgeqoa 34533 exrecfnlem 34542 finxp0 34554 |
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