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Theorem dif20el 6750
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 6747 . . 3  |-  ( A  e.  ( On  \  2o )  <->  ( A  e.  On  /\  1o  e.  A ) )
21simprbi 452 . 2  |-  ( A  e.  ( On  \  2o )  ->  1o  e.  A )
3 0lt1o 6749 . . 3  |-  (/)  e.  1o
4 eldifi 3470 . . . 4  |-  ( A  e.  ( On  \  2o )  ->  A  e.  On )
5 ontr1 4628 . . . 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 659 . 2  |-  ( A  e.  ( On  \  2o )  ->  ( 1o  e.  A  ->  (/)  e.  A
) )
82, 7mpd 15 1  |-  ( A  e.  ( On  \  2o )  ->  (/)  e.  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    e. wcel 1726    \ cdif 3318   (/)c0 3629   Oncon0 4582   1oc1o 6718   2oc2o 6719
This theorem is referenced by:  oeordi  6831  oeworde  6837  oelimcl  6844  oeeulem  6845  oeeui  6846
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-tr 4304  df-eprel 4495  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-suc 4588  df-1o 6725  df-2o 6726
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