Theorem 2onnALT 8657

Description: Shorter proof of 2onn 8656 using Peano's postulates that depends on ax-un 7734. (Contributed by NM, 28-Sep-2004.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
2onnALT	⊢ 2o ∈ ω

Proof of Theorem 2onnALT

Step	Hyp	Ref	Expression
1		df-2o 8481	. 2 ⊢ 2o = suc 1o
2		1onn 8654	. . 3 ⊢ 1o ∈ ω
3		peano2 7890	. . 3 ⊢ (1o ∈ ω → suc 1o ∈ ω)
4	2, 3	ax-mp 5	. 2 ⊢ suc 1o ∈ ω
5	1, 4	eqeltri 2824	1 ⊢ 2o ∈ ω

Syntax hints:   ∈ wcel 2099  suc csuc 6365  ωcom 7864  1_oc1o 8473  2_oc2o 8474

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-tr 5260  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-om 7865  df-1o 8480  df-2o 8481

This theorem is referenced by: (None)