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Theorem ondif2 6746
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 3330 . 2  |-  ( A  e.  ( On  \  2o )  <->  ( A  e.  On  /\  -.  A  e.  2o ) )
2 1on 6731 . . . . 5  |-  1o  e.  On
3 ontri1 4615 . . . . . 6  |-  ( ( A  e.  On  /\  1o  e.  On )  -> 
( A  C_  1o  <->  -.  1o  e.  A ) )
4 onsssuc 4669 . . . . . . 7  |-  ( ( A  e.  On  /\  1o  e.  On )  -> 
( A  C_  1o  <->  A  e.  suc  1o ) )
5 df-2o 6725 . . . . . . . 8  |-  2o  =  suc  1o
65eleq2i 2500 . . . . . . 7  |-  ( A  e.  2o  <->  A  e.  suc  1o )
74, 6syl6bbr 255 . . . . . 6  |-  ( ( A  e.  On  /\  1o  e.  On )  -> 
( A  C_  1o  <->  A  e.  2o ) )
83, 7bitr3d 247 . . . . 5  |-  ( ( A  e.  On  /\  1o  e.  On )  -> 
( -.  1o  e.  A 
<->  A  e.  2o ) )
92, 8mpan2 653 . . . 4  |-  ( A  e.  On  ->  ( -.  1o  e.  A  <->  A  e.  2o ) )
109con1bid 321 . . 3  |-  ( A  e.  On  ->  ( -.  A  e.  2o  <->  1o  e.  A ) )
1110pm5.32i 619 . 2  |-  ( ( A  e.  On  /\  -.  A  e.  2o ) 
<->  ( A  e.  On  /\  1o  e.  A ) )
121, 11bitri 241 1  |-  ( A  e.  ( On  \  2o )  <->  ( A  e.  On  /\  1o  e.  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    /\ wa 359    e. wcel 1725    \ cdif 3317    C_ wss 3320   Oncon0 4581   suc csuc 4583   1oc1o 6717   2oc2o 6718
This theorem is referenced by:  dif20el  6749  oeordi  6830  oewordi  6834  oaabs2  6888  omabs  6890  cnfcom3clem  7662  infxpenc2lem1  7900
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-tr 4303  df-eprel 4494  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-suc 4587  df-1o 6724  df-2o 6725
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