Theorem ondif2 8466

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 3924	. 2 ⊢ (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ ¬ 𝐴 ∈ 2o))
2		1on 8446	. . . . 5 ⊢ 1o ∈ On
3		ontri1 6366	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 1o ∈ On) → (𝐴 ⊆ 1o ↔ ¬ 1o ∈ 𝐴))
4		onsssuc 6424	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 1o ∈ On) → (𝐴 ⊆ 1o ↔ 𝐴 ∈ suc 1o))
5		df-2o 8435	. . . . . . . 8 ⊢ 2o = suc 1o
6	5	eleq2i 2820	. . . . . . 7 ⊢ (𝐴 ∈ 2o ↔ 𝐴 ∈ suc 1o)
7	4, 6	bitr4di 289	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 1o ∈ On) → (𝐴 ⊆ 1o ↔ 𝐴 ∈ 2o))
8	3, 7	bitr3d 281	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 1o ∈ On) → (¬ 1o ∈ 𝐴 ↔ 𝐴 ∈ 2o))
9	2, 8	mpan2 691	. . . 4 ⊢ (𝐴 ∈ On → (¬ 1o ∈ 𝐴 ↔ 𝐴 ∈ 2o))
10	9	con1bid 355	. . 3 ⊢ (𝐴 ∈ On → (¬ 𝐴 ∈ 2o ↔ 1o ∈ 𝐴))
11	10	pm5.32i 574	. 2 ⊢ ((𝐴 ∈ On ∧ ¬ 𝐴 ∈ 2o) ↔ (𝐴 ∈ On ∧ 1o ∈ 𝐴))
12	1, 11	bitri 275	1 ⊢ (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ 1o ∈ 𝐴))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   ∈ wcel 2109   ∖ cdif 3911   ⊆ wss 3914  Oncon0 6332  suc csuc 6334  1_oc1o 8427  2_oc2o 8428

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-tr 5215  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-ord 6335  df-on 6336  df-suc 6338  df-1o 8434  df-2o 8435

This theorem is referenced by:  dif20el  8469  oeordi  8551  oewordi  8555  oaabs2  8613  omabs  8615  cnfcom3clem  9658  infxpenc2lem1  9972  onexoegt  43233  oege2  43296  rp-oelim2  43297  oeord2lim  43298  oeord2i  43299  oeord2com  43300  nnoeomeqom  43301  oenord1  43305  cantnftermord  43309  cantnfresb  43313  cantnf2  43314  omabs2  43321  omcl2  43322