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Theorem ordtoplem 25899
Description: Membership of the class of successor ordinals. (Contributed by Chen-Pang He, 1-Nov-2015.)
Hypothesis
Ref Expression
ordtoplem.1  |-  ( U. A  e.  On  ->  suc  U. A  e.  S
)
Assertion
Ref Expression
ordtoplem  |-  ( Ord 
A  ->  ( A  =/=  U. A  ->  A  e.  S ) )

Proof of Theorem ordtoplem
StepHypRef Expression
1 df-ne 2552 . 2  |-  ( A  =/=  U. A  <->  -.  A  =  U. A )
2 ordeleqon 4709 . . . . . 6  |-  ( Ord 
A  <->  ( A  e.  On  \/  A  =  On ) )
3 unon 4751 . . . . . . . . 9  |-  U. On  =  On
43eqcomi 2391 . . . . . . . 8  |-  On  =  U. On
5 id 20 . . . . . . . 8  |-  ( A  =  On  ->  A  =  On )
6 unieq 3966 . . . . . . . 8  |-  ( A  =  On  ->  U. A  =  U. On )
74, 5, 63eqtr4a 2445 . . . . . . 7  |-  ( A  =  On  ->  A  =  U. A )
87orim2i 505 . . . . . 6  |-  ( ( A  e.  On  \/  A  =  On )  ->  ( A  e.  On  \/  A  =  U. A ) )
92, 8sylbi 188 . . . . 5  |-  ( Ord 
A  ->  ( A  e.  On  \/  A  = 
U. A ) )
109orcomd 378 . . . 4  |-  ( Ord 
A  ->  ( A  =  U. A  \/  A  e.  On ) )
1110ord 367 . . 3  |-  ( Ord 
A  ->  ( -.  A  =  U. A  ->  A  e.  On )
)
12 orduniorsuc 4750 . . . 4  |-  ( Ord 
A  ->  ( A  =  U. A  \/  A  =  suc  U. A ) )
1312ord 367 . . 3  |-  ( Ord 
A  ->  ( -.  A  =  U. A  ->  A  =  suc  U. A
) )
14 onuni 4713 . . . 4  |-  ( A  e.  On  ->  U. A  e.  On )
15 ordtoplem.1 . . . 4  |-  ( U. A  e.  On  ->  suc  U. A  e.  S
)
16 eleq1a 2456 . . . 4  |-  ( suc  U. A  e.  S  ->  ( A  =  suc  U. A  ->  A  e.  S ) )
1714, 15, 163syl 19 . . 3  |-  ( A  e.  On  ->  ( A  =  suc  U. A  ->  A  e.  S ) )
1811, 13, 17syl6c 62 . 2  |-  ( Ord 
A  ->  ( -.  A  =  U. A  ->  A  e.  S )
)
191, 18syl5bi 209 1  |-  ( Ord 
A  ->  ( A  =/=  U. A  ->  A  e.  S ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    = wceq 1649    e. wcel 1717    =/= wne 2550   U.cuni 3957   Ord word 4521   Oncon0 4522   suc csuc 4524
This theorem is referenced by:  ordtop  25900  ordtopcon  25903  ordtopt0  25906
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-tr 4244  df-eprel 4435  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-suc 4528
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