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Theorem ondif2 8540
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 3961 . 2 (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ ¬ 𝐴 ∈ 2o))
2 1on 8518 . . . . 5 1o ∈ On
3 ontri1 6418 . . . . . 6 ((𝐴 ∈ On ∧ 1o ∈ On) → (𝐴 ⊆ 1o ↔ ¬ 1o𝐴))
4 onsssuc 6474 . . . . . . 7 ((𝐴 ∈ On ∧ 1o ∈ On) → (𝐴 ⊆ 1o𝐴 ∈ suc 1o))
5 df-2o 8507 . . . . . . . 8 2o = suc 1o
65eleq2i 2833 . . . . . . 7 (𝐴 ∈ 2o𝐴 ∈ suc 1o)
74, 6bitr4di 289 . . . . . 6 ((𝐴 ∈ On ∧ 1o ∈ On) → (𝐴 ⊆ 1o𝐴 ∈ 2o))
83, 7bitr3d 281 . . . . 5 ((𝐴 ∈ On ∧ 1o ∈ On) → (¬ 1o𝐴𝐴 ∈ 2o))
92, 8mpan2 691 . . . 4 (𝐴 ∈ On → (¬ 1o𝐴𝐴 ∈ 2o))
109con1bid 355 . . 3 (𝐴 ∈ On → (¬ 𝐴 ∈ 2o ↔ 1o𝐴))
1110pm5.32i 574 . 2 ((𝐴 ∈ On ∧ ¬ 𝐴 ∈ 2o) ↔ (𝐴 ∈ On ∧ 1o𝐴))
121, 11bitri 275 1 (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ 1o𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wa 395  wcel 2108  cdif 3948  wss 3951  Oncon0 6384  suc csuc 6386  1oc1o 8499  2oc2o 8500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388  df-suc 6390  df-1o 8506  df-2o 8507
This theorem is referenced by:  dif20el  8543  oeordi  8625  oewordi  8629  oaabs2  8687  omabs  8689  cnfcom3clem  9745  infxpenc2lem1  10059  onexoegt  43256  oege2  43320  rp-oelim2  43321  oeord2lim  43322  oeord2i  43323  oeord2com  43324  nnoeomeqom  43325  oenord1  43329  cantnftermord  43333  cantnfresb  43337  cantnf2  43338  omabs2  43345  omcl2  43346
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