Theorem 1onnALT 8626

Description: Shorter proof of 1onn 8625 using Peano's postulates that depends on ax-un 7732. (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 8452	. 2 ⊢ 1o = suc ∅
2		peano1 7884	. . 3 ⊢ ∅ ∈ ω
3		peano2 7885	. . 3 ⊢ (∅ ∈ ω → suc ∅ ∈ ω)
4	2, 3	ax-mp 5	. 2 ⊢ suc ∅ ∈ ω
5	1, 4	eqeltri 2857	1 ⊢ 1o ∈ ω

Syntax hints:   ∈ wcel 2141  ∅c0 4285  suc csuc 6362  ωcom 7861  1_oc1o 8445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-tr 5218  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-om 7862  df-1o 8452

This theorem is referenced by: (None)