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Theorem ondif2 8116
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 3943 . 2 (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ ¬ 𝐴 ∈ 2o))
2 1on 8098 . . . . 5 1o ∈ On
3 ontri1 6218 . . . . . 6 ((𝐴 ∈ On ∧ 1o ∈ On) → (𝐴 ⊆ 1o ↔ ¬ 1o𝐴))
4 onsssuc 6271 . . . . . . 7 ((𝐴 ∈ On ∧ 1o ∈ On) → (𝐴 ⊆ 1o𝐴 ∈ suc 1o))
5 df-2o 8092 . . . . . . . 8 2o = suc 1o
65eleq2i 2901 . . . . . . 7 (𝐴 ∈ 2o𝐴 ∈ suc 1o)
74, 6syl6bbr 290 . . . . . 6 ((𝐴 ∈ On ∧ 1o ∈ On) → (𝐴 ⊆ 1o𝐴 ∈ 2o))
83, 7bitr3d 282 . . . . 5 ((𝐴 ∈ On ∧ 1o ∈ On) → (¬ 1o𝐴𝐴 ∈ 2o))
92, 8mpan2 687 . . . 4 (𝐴 ∈ On → (¬ 1o𝐴𝐴 ∈ 2o))
109con1bid 357 . . 3 (𝐴 ∈ On → (¬ 𝐴 ∈ 2o ↔ 1o𝐴))
1110pm5.32i 575 . 2 ((𝐴 ∈ On ∧ ¬ 𝐴 ∈ 2o) ↔ (𝐴 ∈ On ∧ 1o𝐴))
121, 11bitri 276 1 (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ 1o𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207  wa 396  wcel 2105  cdif 3930  wss 3933  Oncon0 6184  suc csuc 6186  1oc1o 8084  2oc2o 8085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-tr 5164  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-ord 6187  df-on 6188  df-suc 6190  df-1o 8091  df-2o 8092
This theorem is referenced by:  dif20el  8119  oeordi  8202  oewordi  8206  oaabs2  8261  omabs  8263  cnfcom3clem  9156  infxpenc2lem1  9433
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