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Theorem ondif2 6942
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  e.  ( On  \  2o )  <->  ( A  e.  On  /\  1o  e.  A ) )

Proof of Theorem ondif2
StepHypRef Expression
1 eldif 3338 . 2  |-  ( A  e.  ( On  \  2o )  <->  ( A  e.  On  /\  -.  A  e.  2o ) )
2 1on 6927 . . . . 5  |-  1o  e.  On
3 ontri1 4753 . . . . . 6  |-  ( ( A  e.  On  /\  1o  e.  On )  -> 
( A  C_  1o  <->  -.  1o  e.  A ) )
4 onsssuc 4806 . . . . . . 7  |-  ( ( A  e.  On  /\  1o  e.  On )  -> 
( A  C_  1o  <->  A  e.  suc  1o ) )
5 df-2o 6921 . . . . . . . 8  |-  2o  =  suc  1o
65eleq2i 2507 . . . . . . 7  |-  ( A  e.  2o  <->  A  e.  suc  1o )
74, 6syl6bbr 263 . . . . . 6  |-  ( ( A  e.  On  /\  1o  e.  On )  -> 
( A  C_  1o  <->  A  e.  2o ) )
83, 7bitr3d 255 . . . . 5  |-  ( ( A  e.  On  /\  1o  e.  On )  -> 
( -.  1o  e.  A 
<->  A  e.  2o ) )
92, 8mpan2 671 . . . 4  |-  ( A  e.  On  ->  ( -.  1o  e.  A  <->  A  e.  2o ) )
109con1bid 330 . . 3  |-  ( A  e.  On  ->  ( -.  A  e.  2o  <->  1o  e.  A ) )
1110pm5.32i 637 . 2  |-  ( ( A  e.  On  /\  -.  A  e.  2o ) 
<->  ( A  e.  On  /\  1o  e.  A ) )
121, 11bitri 249 1  |-  ( A  e.  ( On  \  2o )  <->  ( A  e.  On  /\  1o  e.  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 369    e. wcel 1756    \ cdif 3325    C_ wss 3328   Oncon0 4719   suc csuc 4721   1oc1o 6913   2oc2o 6914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-tr 4386  df-eprel 4632  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-suc 4725  df-1o 6920  df-2o 6921
This theorem is referenced by:  dif20el  6945  oeordi  7026  oewordi  7030  oaabs2  7084  omabs  7086  cnfcom3clem  7938  cnfcom3clemOLD  7946  infxpenc2lem1  8185
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