Theorem 1onnALT 8571

Description: Shorter proof of 1onn 8570 using Peano's postulates that depends on ax-un 7681. (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 8399	. 2 ⊢ 1o = suc ∅
2		peano1 7832	. . 3 ⊢ ∅ ∈ ω
3		peano2 7833	. . 3 ⊢ (∅ ∈ ω → suc ∅ ∈ ω)
4	2, 3	ax-mp 5	. 2 ⊢ suc ∅ ∈ ω
5	1, 4	eqeltri 2837	1 ⊢ 1o ∈ ω

Syntax hints:   ∈ wcel 2121  ∅c0 4263  suc csuc 6315  ωcom 7809  1_oc1o 8392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5220  ax-nul 5230  ax-pr 5364  ax-un 7681

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3904  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-opab 5137  df-tr 5182  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-we 5575  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-om 7810  df-1o 8399

This theorem is referenced by: (None)