Theorem peano1 3139

Description: Zero is a natural number. One of Peano's 5 postulates for arithmetic. Proposition 7.30(1) of [TakeutiZaring] p. 42. Note: Unlike most textbooks, our proofs of peano1 3139 through peano5 3143 do not use the Axiom of Infinity. Unlike Takeuti and Zaring, they also do not use the Axiom of Regularity.

Assertion

Ref	Expression
peano1	⊢ ∅ ∈ ω

Proof of Theorem peano1

Step	Hyp	Ref	Expression
1		limom 3136	. 2 ⊢ Lim ω
2		0ellim 3021	. 2 ⊢ (Lim ω → ∅ ∈ ω)
3	1, 2	ax-mp 7	1 ⊢ ∅ ∈ ω

Syntax hints:   ∈ wcel 955  ∅c0 2270  Lim wlim 2939  ωcom 3121

This theorem is referenced by:  fr0t 3937  nnmcl 4214  nnecl 4215  nnmsucr 4224  1onn 4237  nneob 4239  snfi 4413  0sdom1dom 4504  infn0 4512  unblem2 4518  unfilem3 4526  unifi 4532  inf0 4578  infeq5 4593  axinf2 4596  dfom3 4602  noinfep 4612  trcl 4617  cardlim 4823  alephgeom 4854  alephfplem4 4871  mulclpi 4993  1lt2pi 5004  om2uzran 6237  uzrdgini 6240

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122

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