Theorem 2onnALT 8645

Description: Shorter proof of 2onn 8644 using Peano's postulates that depends on ax-un 7749. (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 8470	. 2 ⊢ 2o = suc 1o
2		1onn 8642	. . 3 ⊢ 1o ∈ ω
3		peano2 7899	. . 3 ⊢ (1o ∈ ω → suc 1o ∈ ω)
4	2, 3	ax-mp 5	. 2 ⊢ suc 1o ∈ ω
5	1, 4	eqeltri 2857	1 ⊢ 2o ∈ ω

This proof depends on syntax axioms:   ∈ wcel 2145  suc csuc 6363  ωcom 7875  1_oc1o 8462  2_oc2o 8463

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-tr 5213  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-om 7876  df-1o 8469  df-2o 8470

This theorem is used by: (None)