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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 ∧ 1𝑜𝐴))

Proof of Theorem ondif2
StepHypRef 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 ↔ ¬ 1𝑜𝐴))
4 onsssuc 5851 . . . . . . 7 ((𝐴 ∈ On ∧ 1𝑜 ∈ On) → (𝐴 ⊆ 1𝑜𝐴 ∈ suc 1𝑜))
5 df-2o 7606 . . . . . . . 8 2𝑜 = suc 1𝑜
65eleq2i 2722 . . . . . . 7 (𝐴 ∈ 2𝑜𝐴 ∈ suc 1𝑜)
74, 6syl6bbr 278 . . . . . 6 ((𝐴 ∈ On ∧ 1𝑜 ∈ On) → (𝐴 ⊆ 1𝑜𝐴 ∈ 2𝑜))
83, 7bitr3d 270 . . . . 5 ((𝐴 ∈ On ∧ 1𝑜 ∈ On) → (¬ 1𝑜𝐴𝐴 ∈ 2𝑜))
92, 8mpan2 707 . . . 4 (𝐴 ∈ On → (¬ 1𝑜𝐴𝐴 ∈ 2𝑜))
109con1bid 344 . . 3 (𝐴 ∈ On → (¬ 𝐴 ∈ 2𝑜 ↔ 1𝑜𝐴))
1110pm5.32i 670 . 2 ((𝐴 ∈ On ∧ ¬ 𝐴 ∈ 2𝑜) ↔ (𝐴 ∈ On ∧ 1𝑜𝐴))
121, 11bitri 264 1 (𝐴 ∈ (On ∖ 2𝑜) ↔ (𝐴 ∈ On ∧ 1𝑜𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 383  wcel 2030  cdif 3604  wss 3607  Oncon0 5761  suc csuc 5763  1𝑜c1o 7598  2𝑜c2o 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
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