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Theorem ondif2 7164
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 3491 . 2  |-  ( A  e.  ( On  \  2o )  <->  ( A  e.  On  /\  -.  A  e.  2o ) )
2 1on 7149 . . . . 5  |-  1o  e.  On
3 ontri1 4918 . . . . . 6  |-  ( ( A  e.  On  /\  1o  e.  On )  -> 
( A  C_  1o  <->  -.  1o  e.  A ) )
4 onsssuc 4971 . . . . . . 7  |-  ( ( A  e.  On  /\  1o  e.  On )  -> 
( A  C_  1o  <->  A  e.  suc  1o ) )
5 df-2o 7143 . . . . . . . 8  |-  2o  =  suc  1o
65eleq2i 2545 . . . . . . 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 1767    \ cdif 3478    C_ wss 3481   Oncon0 4884   suc csuc 4886   1oc1o 7135   2oc2o 7136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-tr 4547  df-eprel 4797  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-suc 4890  df-1o 7142  df-2o 7143
This theorem is referenced by:  dif20el  7167  oeordi  7248  oewordi  7252  oaabs2  7306  omabs  7308  cnfcom3clem  8161  cnfcom3clemOLD  8169  infxpenc2lem1  8408
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