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Theorem ordtoplem 24874
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 2448 . 2  |-  ( A  =/=  U. A  <->  -.  A  =  U. A )
2 ordeleqon 4580 . . . . . 6  |-  ( Ord 
A  <->  ( A  e.  On  \/  A  =  On ) )
3 unon 4622 . . . . . . . . 9  |-  U. On  =  On
43eqcomi 2287 . . . . . . . 8  |-  On  =  U. On
5 id 19 . . . . . . . 8  |-  ( A  =  On  ->  A  =  On )
6 unieq 3836 . . . . . . . 8  |-  ( A  =  On  ->  U. A  =  U. On )
74, 5, 63eqtr4a 2341 . . . . . . 7  |-  ( A  =  On  ->  A  =  U. A )
87orim2i 504 . . . . . 6  |-  ( ( A  e.  On  \/  A  =  On )  ->  ( A  e.  On  \/  A  =  U. A ) )
92, 8sylbi 187 . . . . 5  |-  ( Ord 
A  ->  ( A  e.  On  \/  A  = 
U. A ) )
109orcomd 377 . . . 4  |-  ( Ord 
A  ->  ( A  =  U. A  \/  A  e.  On ) )
1110ord 366 . . 3  |-  ( Ord 
A  ->  ( -.  A  =  U. A  ->  A  e.  On )
)
12 orduniorsuc 4621 . . . 4  |-  ( Ord 
A  ->  ( A  =  U. A  \/  A  =  suc  U. A ) )
1312ord 366 . . 3  |-  ( Ord 
A  ->  ( -.  A  =  U. A  ->  A  =  suc  U. A
) )
14 onuni 4584 . . . 4  |-  ( A  e.  On  ->  U. A  e.  On )
15 ordtoplem.1 . . . 4  |-  ( U. A  e.  On  ->  suc  U. A  e.  S
)
16 eleq1a 2352 . . . 4  |-  ( suc  U. A  e.  S  ->  ( A  =  suc  U. A  ->  A  e.  S ) )
1714, 15, 163syl 18 . . 3  |-  ( A  e.  On  ->  ( A  =  suc  U. A  ->  A  e.  S ) )
1811, 13, 17syl6c 60 . 2  |-  ( Ord 
A  ->  ( -.  A  =  U. A  ->  A  e.  S )
)
191, 18syl5bi 208 1  |-  ( Ord 
A  ->  ( A  =/=  U. A  ->  A  e.  S ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    = wceq 1623    e. wcel 1684    =/= wne 2446   U.cuni 3827   Ord word 4391   Oncon0 4392   suc csuc 4394
This theorem is referenced by:  ordtop  24875  ordtopcon  24878  ordtopt0  24881
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398
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