Theorem peano1 3144

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 3144 through peano5 3148 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 3141	. 2 ⊢ Lim ω
2		0ellim 3026	. 2 ⊢ (Lim ω → ∅ ∈ ω)
3	1, 2	ax-mp 7	1 ⊢ ∅ ∈ ω

Syntax hints:   ∈ wcel 956  ∅c0 2276  Lim wlim 2944  ωcom 3126

This theorem is referenced by:  fr0t 3943  nnmcl 4220  nnecl 4221  nnmsucr 4230  1onn 4243  nneob 4245  snfi 4419  0sdom1dom 4510  infn0 4518  unblem2 4524  unfilem3 4532  unifi 4538  inf0 4586  infeq5 4601  axinf2 4604  dfom3 4610  noinfep 4620  trcl 4625  cardlim 4831  alephgeom 4862  alephfplem4 4879  mulclpi 5001  1lt2pi 5012  om2uzran 6245  uzrdgini 6248

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127

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