Theorem 2onnALT 8699

Description: Shorter proof of 2onn 8698 using Peano's postulates that depends on ax-un 7770. (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 8523	. 2 ⊢ 2o = suc 1o
2		1onn 8696	. . 3 ⊢ 1o ∈ ω
3		peano2 7929	. . 3 ⊢ (1o ∈ ω → suc 1o ∈ ω)
4	2, 3	ax-mp 5	. 2 ⊢ suc 1o ∈ ω
5	1, 4	eqeltri 2840	1 ⊢ 2o ∈ ω

Syntax hints:   ∈ wcel 2108  suc csuc 6397  ωcom 7903  1_oc1o 8515  2_oc2o 8516

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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-tr 5284  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-om 7904  df-1o 8522  df-2o 8523

This theorem is referenced by: (None)