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Theorem peano1 6954
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 6954 through peano5 6958 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.)
Assertion
Ref Expression
peano1 ∅ ∈ ω

Proof of Theorem peano1
StepHypRef Expression
1 limom 6949 . 2 Lim ω
2 0ellim 5690 . 2 (Lim ω → ∅ ∈ ω)
31, 2ax-mp 5 1 ∅ ∈ ω
Colors of variables: wff setvar class
Syntax hints:  wcel 1976  c0 3873  Lim wlim 5627  ωcom 6934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-tr 4675  df-eprel 4939  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-om 6935
This theorem is referenced by:  onnseq  7305  rdg0  7381  fr0g  7395  seqomlem3  7411  oa1suc  7475  om1  7486  oe1  7488  nna0r  7553  nnm0r  7554  nnmcl  7556  nnecl  7557  nnmsucr  7569  nnaword1  7573  nnaordex  7582  1onn  7583  oaabs2  7589  nnm1  7592  nneob  7596  omopth  7602  snfi  7900  0sdom1dom  8020  0fin  8050  findcard2  8062  nnunifi  8073  unblem2  8075  infn0  8084  unfilem3  8088  dffi3  8197  inf0  8378  infeq5i  8393  axinf2  8397  dfom3  8404  infdifsn  8414  noinfep  8417  cantnflt  8429  cnfcomlem  8456  cnfcom  8457  cnfcom2lem  8458  cnfcom3lem  8460  cnfcom3  8461  trcl  8464  rankdmr1  8524  rankeq0b  8583  cardlim  8658  infxpenc  8701  infxpenc2  8705  alephgeom  8765  alephfplem4  8790  ackbij1lem13  8914  ackbij1  8920  ackbij1b  8921  ominf4  8994  fin23lem16  9017  fin23lem31  9025  fin23lem40  9033  isf32lem9  9043  isf34lem7  9061  isf34lem6  9062  fin1a2lem6  9087  fin1a2lem7  9088  fin1a2lem11  9092  axdc3lem2  9133  axdc3lem4  9135  axdc4lem  9137  axcclem  9139  axdclem2  9202  pwfseqlem5  9341  omina  9369  wunex3  9419  1lt2pi  9583  1nn  10878  om2uzrani  12568  uzrdg0i  12575  fzennn  12584  axdc4uzlem  12599  hash1  13005  ltbwe  19239  2ndcdisj2  21012  snct  28680  trpredpred  30778  0hf  31260  neibastop2lem  31331  rdgeqoa  32197  finxp0  32207
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