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Theorem ondif2 8498
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 3951 . 2 (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ ¬ 𝐴 ∈ 2o))
2 1on 8474 . . . . 5 1o ∈ On
3 ontri1 6389 . . . . . 6 ((𝐴 ∈ On ∧ 1o ∈ On) → (𝐴 ⊆ 1o ↔ ¬ 1o𝐴))
4 onsssuc 6445 . . . . . . 7 ((𝐴 ∈ On ∧ 1o ∈ On) → (𝐴 ⊆ 1o𝐴 ∈ suc 1o))
5 df-2o 8463 . . . . . . . 8 2o = suc 1o
65eleq2i 2817 . . . . . . 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 688 . . . 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 205  wa 395  wcel 2098  cdif 3938  wss 3941  Oncon0 6355  suc csuc 6357  1oc1o 8455  2oc2o 8456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-tr 5257  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-ord 6358  df-on 6359  df-suc 6361  df-1o 8462  df-2o 8463
This theorem is referenced by:  dif20el  8501  oeordi  8583  oewordi  8587  oaabs2  8645  omabs  8647  cnfcom3clem  9697  infxpenc2lem1  10011  onexoegt  42543  oege2  42607  rp-oelim2  42608  oeord2lim  42609  oeord2i  42610  oeord2com  42611  nnoeomeqom  42612  oenord1  42616  cantnftermord  42620  cantnfresb  42624  cantnf2  42625  omabs2  42632  omcl2  42633
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