Theorem 1onnALT 8613

Description: Shorter proof of 1onn 8612 using Peano's postulates that depends on ax-un 7720. (Contributed by NM, 29-Oct-1995.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
1onnALT	⊢ 1o ∈ ω

Proof of Theorem 1onnALT

Step	Hyp	Ref	Expression
1		df-1o 8439	. 2 ⊢ 1o = suc ∅
2		peano1 7871	. . 3 ⊢ ∅ ∈ ω
3		peano2 7872	. . 3 ⊢ (∅ ∈ ω → suc ∅ ∈ ω)
4	2, 3	ax-mp 5	. 2 ⊢ suc ∅ ∈ ω
5	1, 4	eqeltri 2860	1 ⊢ 1o ∈ ω

Syntax hints:   ∈ wcel 2144  ∅c0 4287  suc csuc 6350  ωcom 7848  1_oc1o 8432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-tr 5210  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-om 7849  df-1o 8439

This theorem is referenced by: (None)