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Theorem ondif2 8452
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 3924 . 2 (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ ¬ 𝐴 ∈ 2o))
2 1on 8428 . . . . 5 1o ∈ On
3 ontri1 6355 . . . . . 6 ((𝐴 ∈ On ∧ 1o ∈ On) → (𝐴 ⊆ 1o ↔ ¬ 1o𝐴))
4 onsssuc 6411 . . . . . . 7 ((𝐴 ∈ On ∧ 1o ∈ On) → (𝐴 ⊆ 1o𝐴 ∈ suc 1o))
5 df-2o 8417 . . . . . . . 8 2o = suc 1o
65eleq2i 2826 . . . . . . 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 690 . . . 4 (𝐴 ∈ On → (¬ 1o𝐴𝐴 ∈ 2o))
109con1bid 356 . . 3 (𝐴 ∈ On → (¬ 𝐴 ∈ 2o ↔ 1o𝐴))
1110pm5.32i 576 . 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 397  wcel 2107  cdif 3911  wss 3914  Oncon0 6321  suc csuc 6323  1oc1o 8409  2oc2o 8410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-tr 5227  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-ord 6324  df-on 6325  df-suc 6327  df-1o 8416  df-2o 8417
This theorem is referenced by:  dif20el  8455  oeordi  8538  oewordi  8542  oaabs2  8599  omabs  8601  cnfcom3clem  9649  infxpenc2lem1  9963  onexoegt  41625  oege2  41689  rp-oelim2  41690  oeord2lim  41691  oeord2i  41692  oeord2com  41693  nnoeomeqom  41694  oenord1  41698  cantnftermord  41702  cantnfresb  41706  cantnf2  41707  omabs2  41714  omcl2  41715
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