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Theorem ondif2 6501
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 3162 . 2  |-  ( A  e.  ( On  \  2o )  <->  ( A  e.  On  /\  -.  A  e.  2o ) )
2 1on 6486 . . . . 5  |-  1o  e.  On
3 ontri1 4426 . . . . . 6  |-  ( ( A  e.  On  /\  1o  e.  On )  -> 
( A  C_  1o  <->  -.  1o  e.  A ) )
4 onsssuc 4480 . . . . . . 7  |-  ( ( A  e.  On  /\  1o  e.  On )  -> 
( A  C_  1o  <->  A  e.  suc  1o ) )
5 df-2o 6480 . . . . . . . 8  |-  2o  =  suc  1o
65eleq2i 2347 . . . . . . 7  |-  ( A  e.  2o  <->  A  e.  suc  1o )
74, 6syl6bbr 254 . . . . . 6  |-  ( ( A  e.  On  /\  1o  e.  On )  -> 
( A  C_  1o  <->  A  e.  2o ) )
83, 7bitr3d 246 . . . . 5  |-  ( ( A  e.  On  /\  1o  e.  On )  -> 
( -.  1o  e.  A 
<->  A  e.  2o ) )
92, 8mpan2 652 . . . 4  |-  ( A  e.  On  ->  ( -.  1o  e.  A  <->  A  e.  2o ) )
109con1bid 320 . . 3  |-  ( A  e.  On  ->  ( -.  A  e.  2o  <->  1o  e.  A ) )
1110pm5.32i 618 . 2  |-  ( ( A  e.  On  /\  -.  A  e.  2o ) 
<->  ( A  e.  On  /\  1o  e.  A ) )
121, 11bitri 240 1  |-  ( A  e.  ( On  \  2o )  <->  ( A  e.  On  /\  1o  e.  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    /\ wa 358    e. wcel 1684    \ cdif 3149    C_ wss 3152   Oncon0 4392   suc csuc 4394   1oc1o 6472   2oc2o 6473
This theorem is referenced by:  dif20el  6504  oeordi  6585  oewordi  6589  oaabs2  6643  omabs  6645  cnfcom3clem  7408  infxpenc2lem1  7646
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398  df-1o 6479  df-2o 6480
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