Theorem ondif2 7582

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 3584	. 2 ⊢ (𝐴 ∈ (On ∖ 2𝑜) ↔ (𝐴 ∈ On ∧ ¬ 𝐴 ∈ 2𝑜))
2		1on 7567	. . . . 5 ⊢ 1𝑜 ∈ On
3		ontri1 5757	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 1𝑜 ∈ On) → (𝐴 ⊆ 1𝑜 ↔ ¬ 1𝑜 ∈ 𝐴))
4		onsssuc 5813	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 1𝑜 ∈ On) → (𝐴 ⊆ 1𝑜 ↔ 𝐴 ∈ suc 1𝑜))
5		df-2o 7561	. . . . . . . 8 ⊢ 2𝑜 = suc 1𝑜
6	5	eleq2i 2693	. . . . . . 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 345	. . 3 ⊢ (𝐴 ∈ On → (¬ 𝐴 ∈ 2𝑜 ↔ 1𝑜 ∈ 𝐴))
11	10	pm5.32i 669	. 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 384   ∈ wcel 1990   ∖ cdif 3571   ⊆ wss 3574  Oncon0 5723  suc csuc 5725  1_𝑜c1o 7553  2_𝑜c2o 7554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-suc 5729  df-1o 7560  df-2o 7561

This theorem is referenced by:  dif20el  7585  oeordi  7667  oewordi  7671  oaabs2  7725  omabs  7727  cnfcom3clem  8602  infxpenc2lem1  8842