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Theorem dif20el 7167
Description: An ordinal greater than one is greater than zero. (Contributed by Mario Carneiro, 25-May-2015.)
Assertion
Ref Expression
dif20el  |-  ( A  e.  ( On  \  2o )  ->  (/)  e.  A
)

Proof of Theorem dif20el
StepHypRef Expression
1 ondif2 7164 . . 3  |-  ( A  e.  ( On  \  2o )  <->  ( A  e.  On  /\  1o  e.  A ) )
21simprbi 464 . 2  |-  ( A  e.  ( On  \  2o )  ->  1o  e.  A )
3 0lt1o 7166 . . 3  |-  (/)  e.  1o
4 eldifi 3631 . . . 4  |-  ( A  e.  ( On  \  2o )  ->  A  e.  On )
5 ontr1 4930 . . . 4  |-  ( A  e.  On  ->  (
( (/)  e.  1o  /\  1o  e.  A )  ->  (/) 
e.  A ) )
64, 5syl 16 . . 3  |-  ( A  e.  ( On  \  2o )  ->  ( (
(/)  e.  1o  /\  1o  e.  A )  ->  (/)  e.  A
) )
73, 6mpani 676 . 2  |-  ( A  e.  ( On  \  2o )  ->  ( 1o  e.  A  ->  (/)  e.  A
) )
82, 7mpd 15 1  |-  ( A  e.  ( On  \  2o )  ->  (/)  e.  A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1767    \ cdif 3478   (/)c0 3790   Oncon0 4884   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:  oeordi  7248  oeworde  7254  oelimcl  7261  oeeulem  7262  oeeui  7263
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