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Theorem dif20el 6937
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 6934 . . 3  |-  ( A  e.  ( On  \  2o )  <->  ( A  e.  On  /\  1o  e.  A ) )
21simprbi 464 . 2  |-  ( A  e.  ( On  \  2o )  ->  1o  e.  A )
3 0lt1o 6936 . . 3  |-  (/)  e.  1o
4 eldifi 3473 . . . 4  |-  ( A  e.  ( On  \  2o )  ->  A  e.  On )
5 ontr1 4760 . . . 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 1756    \ cdif 3320   (/)c0 3632   Oncon0 4714   1oc1o 6905   2oc2o 6906
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-tr 4381  df-eprel 4627  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-suc 4720  df-1o 6912  df-2o 6913
This theorem is referenced by:  oeordi  7018  oeworde  7024  oelimcl  7031  oeeulem  7032  oeeui  7033
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