Theorem 2onnALT 8553

Description: Shorter proof of 2onn 8552 using Peano's postulates that depends on ax-un 7663. (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 8381	. 2 ⊢ 2o = suc 1o
2		1onn 8550	. . 3 ⊢ 1o ∈ ω
3		peano2 7815	. . 3 ⊢ (1o ∈ ω → suc 1o ∈ ω)
4	2, 3	ax-mp 5	. 2 ⊢ suc 1o ∈ ω
5	1, 4	eqeltri 2827	1 ⊢ 2o ∈ ω

Syntax hints:   ∈ wcel 2111  suc csuc 6303  ωcom 7791  1_oc1o 8373  2_oc2o 8374

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-tr 5194  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-om 7792  df-1o 8380  df-2o 8381

This theorem is referenced by: (None)