Theorem ondif2 7469

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 3550	. 2 ⊢ (𝐴 ∈ (On ∖ 2𝑜) ↔ (𝐴 ∈ On ∧ ¬ 𝐴 ∈ 2𝑜))
2		1on 7454	. . . . 5 ⊢ 1𝑜 ∈ On
3		ontri1 5674	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 1𝑜 ∈ On) → (𝐴 ⊆ 1𝑜 ↔ ¬ 1𝑜 ∈ 𝐴))
4		onsssuc 5730	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 1𝑜 ∈ On) → (𝐴 ⊆ 1𝑜 ↔ 𝐴 ∈ suc 1𝑜))
5		df-2o 7448	. . . . . . . 8 ⊢ 2𝑜 = suc 1𝑜
6	5	eleq2i 2680	. . . . . . 7 ⊢ (𝐴 ∈ 2𝑜 ↔ 𝐴 ∈ suc 1𝑜)
7	4, 6	syl6bbr 277	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 1𝑜 ∈ On) → (𝐴 ⊆ 1𝑜 ↔ 𝐴 ∈ 2𝑜))
8	3, 7	bitr3d 269	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 1𝑜 ∈ On) → (¬ 1𝑜 ∈ 𝐴 ↔ 𝐴 ∈ 2𝑜))
9	2, 8	mpan2 703	. . . 4 ⊢ (𝐴 ∈ On → (¬ 1𝑜 ∈ 𝐴 ↔ 𝐴 ∈ 2𝑜))
10	9	con1bid 344	. . 3 ⊢ (𝐴 ∈ On → (¬ 𝐴 ∈ 2𝑜 ↔ 1𝑜 ∈ 𝐴))
11	10	pm5.32i 667	. 2 ⊢ ((𝐴 ∈ On ∧ ¬ 𝐴 ∈ 2𝑜) ↔ (𝐴 ∈ On ∧ 1𝑜 ∈ 𝐴))
12	1, 11	bitri 263	1 ⊢ (𝐴 ∈ (On ∖ 2𝑜) ↔ (𝐴 ∈ On ∧ 1𝑜 ∈ 𝐴))

Syntax hints:  ¬ wn 3   ↔ wb 195   ∧ wa 383   ∈ wcel 1977   ∖ cdif 3537   ⊆ wss 3540  Oncon0 5640  suc csuc 5642  1_𝑜c1o 7440  2_𝑜c2o 7441

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644  df-suc 5646  df-1o 7447  df-2o 7448

This theorem is referenced by:  dif20el  7472  oeordi  7554  oewordi  7558  oaabs2  7612  omabs  7614  cnfcom3clem  8485  infxpenc2lem1  8725