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Theorem ordtoplem 30084
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 2654 . 2  |-  ( A  =/=  U. A  <->  -.  A  =  U. A )
2 ordeleqon 6623 . . . . . 6  |-  ( Ord 
A  <->  ( A  e.  On  \/  A  =  On ) )
3 unon 6665 . . . . . . . . 9  |-  U. On  =  On
43eqcomi 2470 . . . . . . . 8  |-  On  =  U. On
5 id 22 . . . . . . . 8  |-  ( A  =  On  ->  A  =  On )
6 unieq 4259 . . . . . . . 8  |-  ( A  =  On  ->  U. A  =  U. On )
74, 5, 63eqtr4a 2524 . . . . . . 7  |-  ( A  =  On  ->  A  =  U. A )
87orim2i 518 . . . . . 6  |-  ( ( A  e.  On  \/  A  =  On )  ->  ( A  e.  On  \/  A  =  U. A ) )
92, 8sylbi 195 . . . . 5  |-  ( Ord 
A  ->  ( A  e.  On  \/  A  = 
U. A ) )
109orcomd 388 . . . 4  |-  ( Ord 
A  ->  ( A  =  U. A  \/  A  e.  On ) )
1110ord 377 . . 3  |-  ( Ord 
A  ->  ( -.  A  =  U. A  ->  A  e.  On )
)
12 orduniorsuc 6664 . . . 4  |-  ( Ord 
A  ->  ( A  =  U. A  \/  A  =  suc  U. A ) )
1312ord 377 . . 3  |-  ( Ord 
A  ->  ( -.  A  =  U. A  ->  A  =  suc  U. A
) )
14 onuni 6627 . . . 4  |-  ( A  e.  On  ->  U. A  e.  On )
15 ordtoplem.1 . . . 4  |-  ( U. A  e.  On  ->  suc  U. A  e.  S
)
16 eleq1a 2540 . . . 4  |-  ( suc  U. A  e.  S  ->  ( A  =  suc  U. A  ->  A  e.  S ) )
1714, 15, 163syl 20 . . 3  |-  ( A  e.  On  ->  ( A  =  suc  U. A  ->  A  e.  S ) )
1811, 13, 17syl6c 64 . 2  |-  ( Ord 
A  ->  ( -.  A  =  U. A  ->  A  e.  S )
)
191, 18syl5bi 217 1  |-  ( Ord 
A  ->  ( A  =/=  U. A  ->  A  e.  S ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    = wceq 1395    e. wcel 1819    =/= wne 2652   U.cuni 4251   Ord word 4886   Oncon0 4887   suc csuc 4889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-tr 4551  df-eprel 4800  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-suc 4893
This theorem is referenced by:  ordtop  30085  ordtopcon  30088  ordtopt0  30091
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