Theorem 1onnALT 8566

Description: Shorter proof of 1onn 8565 using Peano's postulates that depends on ax-un 7678. (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 8394	. 2 ⊢ 1o = suc ∅
2		peano1 7829	. . 3 ⊢ ∅ ∈ ω
3		peano2 7830	. . 3 ⊢ (∅ ∈ ω → suc ∅ ∈ ω)
4	2, 3	ax-mp 5	. 2 ⊢ suc ∅ ∈ ω
5	1, 4	eqeltri 2831	1 ⊢ 1o ∈ ω

Syntax hints:   ∈ wcel 2114  ∅c0 4263  suc csuc 6314  ωcom 7806  1_oc1o 8387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707  ax-sep 5220  ax-nul 5230  ax-pr 5364  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-ne 2931  df-ral 3050  df-rex 3060  df-rab 3388  df-v 3429  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  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 6315  df-on 6316  df-lim 6317  df-suc 6318  df-om 7807  df-1o 8394

This theorem is referenced by: (None)