Theorem ondif2 7627

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 3617	. 2 ⊢ (𝐴 ∈ (On ∖ 2𝑜) ↔ (𝐴 ∈ On ∧ ¬ 𝐴 ∈ 2𝑜))
2		1on 7612	. . . . 5 ⊢ 1𝑜 ∈ On
3		ontri1 5795	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 1𝑜 ∈ On) → (𝐴 ⊆ 1𝑜 ↔ ¬ 1𝑜 ∈ 𝐴))
4		onsssuc 5851	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 1𝑜 ∈ On) → (𝐴 ⊆ 1𝑜 ↔ 𝐴 ∈ suc 1𝑜))
5		df-2o 7606	. . . . . . . 8 ⊢ 2𝑜 = suc 1𝑜
6	5	eleq2i 2722	. . . . . . 7 ⊢ (𝐴 ∈ 2𝑜 ↔ 𝐴 ∈ suc 1𝑜)
7	4, 6	syl6bbr 278	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 1𝑜 ∈ On) → (𝐴 ⊆ 1𝑜 ↔ 𝐴 ∈ 2𝑜))
8	3, 7	bitr3d 270	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 1𝑜 ∈ On) → (¬ 1𝑜 ∈ 𝐴 ↔ 𝐴 ∈ 2𝑜))
9	2, 8	mpan2 707	. . . 4 ⊢ (𝐴 ∈ On → (¬ 1𝑜 ∈ 𝐴 ↔ 𝐴 ∈ 2𝑜))
10	9	con1bid 344	. . 3 ⊢ (𝐴 ∈ On → (¬ 𝐴 ∈ 2𝑜 ↔ 1𝑜 ∈ 𝐴))
11	10	pm5.32i 670	. 2 ⊢ ((𝐴 ∈ On ∧ ¬ 𝐴 ∈ 2𝑜) ↔ (𝐴 ∈ On ∧ 1𝑜 ∈ 𝐴))
12	1, 11	bitri 264	1 ⊢ (𝐴 ∈ (On ∖ 2𝑜) ↔ (𝐴 ∈ On ∧ 1𝑜 ∈ 𝐴))

Syntax hints:  ¬ wn 3   ↔ wb 196   ∧ wa 383   ∈ wcel 2030   ∖ cdif 3604   ⊆ wss 3607  Oncon0 5761  suc csuc 5763  1_𝑜c1o 7598  2_𝑜c2o 7599

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765  df-suc 5767  df-1o 7605  df-2o 7606

This theorem is referenced by:  dif20el  7630  oeordi  7712  oewordi  7716  oaabs2  7770  omabs  7772  cnfcom3clem  8640  infxpenc2lem1  8880