Theorem 1onnALT 8684

Description: Shorter proof of 1onn 8683 using Peano's postulates that depends on ax-un 7758. (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 8511	. 2 ⊢ 1o = suc ∅
2		peano1 7915	. . 3 ⊢ ∅ ∈ ω
3		peano2 7917	. . 3 ⊢ (∅ ∈ ω → suc ∅ ∈ ω)
4	2, 3	ax-mp 5	. 2 ⊢ suc ∅ ∈ ω
5	1, 4	eqeltri 2836	1 ⊢ 1o ∈ ω

Syntax hints:   ∈ wcel 2107  ∅c0 4340  suc csuc 6391  ωcom 7891  1_oc1o 8504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5303  ax-nul 5313  ax-pr 5439  ax-un 7758

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1541  df-fal 1551  df-ex 1778  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3435  df-v 3481  df-dif 3967  df-un 3969  df-in 3971  df-ss 3981  df-pss 3984  df-nul 4341  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4914  df-br 5150  df-opab 5212  df-tr 5267  df-eprel 5590  df-po 5598  df-so 5599  df-fr 5642  df-we 5644  df-ord 6392  df-on 6393  df-lim 6394  df-suc 6395  df-om 7892  df-1o 8511

This theorem is referenced by: (None)