Theorem ondif2 8427

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 3909	. 2 ⊢ (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ ¬ 𝐴 ∈ 2o))
2		1on 8407	. . . . 5 ⊢ 1o ∈ On
3		ontri1 6349	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 1o ∈ On) → (𝐴 ⊆ 1o ↔ ¬ 1o ∈ 𝐴))
4		onsssuc 6407	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 1o ∈ On) → (𝐴 ⊆ 1o ↔ 𝐴 ∈ suc 1o))
5		df-2o 8396	. . . . . . . 8 ⊢ 2o = suc 1o
6	5	eleq2i 2826	. . . . . . 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 2113   ∖ cdif 3896   ⊆ wss 3899  Oncon0 6315  suc csuc 6317  1_oc1o 8388  2_oc2o 8389

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 2115  ax-9 2123  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

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 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-tr 5204  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-ord 6318  df-on 6319  df-suc 6321  df-1o 8395  df-2o 8396

This theorem is referenced by:  dif20el  8430  oeordi  8513  oewordi  8517  oaabs2  8575  omabs  8577  cnfcom3clem  9612  infxpenc2lem1  9927  onexoegt  43428  oege2  43491  rp-oelim2  43492  oeord2lim  43493  oeord2i  43494  oeord2com  43495  nnoeomeqom  43496  oenord1  43500  cantnftermord  43504  cantnfresb  43508  cantnf2  43509  omabs2  43516  omcl2  43517