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Theorem ondif2 7191
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 3426 . 2  |-  ( A  e.  ( On  \  2o )  <->  ( A  e.  On  /\  -.  A  e.  2o ) )
2 1on 7176 . . . . 5  |-  1o  e.  On
3 ontri1 5446 . . . . . 6  |-  ( ( A  e.  On  /\  1o  e.  On )  -> 
( A  C_  1o  <->  -.  1o  e.  A ) )
4 onsssuc 5499 . . . . . . 7  |-  ( ( A  e.  On  /\  1o  e.  On )  -> 
( A  C_  1o  <->  A  e.  suc  1o ) )
5 df-2o 7170 . . . . . . . 8  |-  2o  =  suc  1o
65eleq2i 2482 . . . . . . 7  |-  ( A  e.  2o  <->  A  e.  suc  1o )
74, 6syl6bbr 265 . . . . . 6  |-  ( ( A  e.  On  /\  1o  e.  On )  -> 
( A  C_  1o  <->  A  e.  2o ) )
83, 7bitr3d 257 . . . . 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 251 1  |-  ( A  e.  ( On  \  2o )  <->  ( A  e.  On  /\  1o  e.  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 186    /\ wa 369    e. wcel 1844    \ cdif 3413    C_ wss 3416   Oncon0 5412   suc csuc 5414   1oc1o 7162   2oc2o 7163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-br 4398  df-opab 4456  df-tr 4492  df-eprel 4736  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-ord 5415  df-on 5416  df-suc 5418  df-1o 7169  df-2o 7170
This theorem is referenced by:  dif20el  7194  oeordi  7275  oewordi  7279  oaabs2  7333  omabs  7335  cnfcom3clem  8183  cnfcom3clemOLD  8191  infxpenc2lem1  8430
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