Theorem ondif2 8548

Description: Two ways to say that 𝐴 is an ordinal greater than one. (Contributed by Mario Carneiro, 21-May-2015.)

Assertion

Ref	Expression
ondif2	⊢ (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ 1o ∈ 𝐴))

Proof of Theorem ondif2

Step	Hyp	Ref	Expression
1		eldif 4052	. 2 ⊢ (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ ¬ 𝐴 ∈ 2o))
2		1on 8527	. . . . 5 ⊢ 1o ∈ On
3		ontri1 6423	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 1o ∈ On) → (𝐴 ⊆ 1o ↔ ¬ 1o ∈ 𝐴))
4		onsssuc 6481	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 1o ∈ On) → (𝐴 ⊆ 1o ↔ 𝐴 ∈ suc 1o))
5		df-2o 8515	. . . . . . . 8 ⊢ 2o = suc 1o
6	5	eleq2i 2856	. . . . . . 7 ⊢ (𝐴 ∈ 2o ↔ 𝐴 ∈ suc 1o)
7	4, 6	bitr4di 292	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 1o ∈ On) → (𝐴 ⊆ 1o ↔ 𝐴 ∈ 2o))
8	3, 7	bitr3d 284	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 1o ∈ On) → (¬ 1o ∈ 𝐴 ↔ 𝐴 ∈ 2o))
9	2, 8	mpan2 704	. . . 4 ⊢ (𝐴 ∈ On → (¬ 1o ∈ 𝐴 ↔ 𝐴 ∈ 2o))
10	9	con1bid 358	. . 3 ⊢ (𝐴 ∈ On → (¬ 𝐴 ∈ 2o ↔ 1o ∈ 𝐴))
11	10	pm5.32i 585	. 2 ⊢ ((𝐴 ∈ On ∧ ¬ 𝐴 ∈ 2o) ↔ (𝐴 ∈ On ∧ 1o ∈ 𝐴))
12	1, 11	bitri 278	1 ⊢ (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ 1o ∈ 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   ∈ wcel 2147   ⊆ wss 3900   ∖ cdif 4048  Oncon0 6388  suc csuc 6390  1_oc1o 8507  2_oc2o 8508

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-tr 5219  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-ord 6391  df-on 6392  df-suc 6394  df-1o 8514  df-2o 8515

This theorem is used by:  dif20el  8551  oeordi  8634  oewordi  8638  oaabs2  8696  omabs  8698  cnfcom3clem  9743  infxpenc2lem1  10137  onexoegt  44507  oege2  44570  rp-oelim2  44571  oeord2lim  44572  oeord2i  44573  oeord2com  44574  nnoeomeqom  44575  oenord1  44579  cantnftermord  44583  cantnfresb  44587  cantnf2  44588  omabs2  44595  omcl2  44596