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Theorem ordtoplem 24946
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 2461 . 2  |-  ( A  =/=  U. A  <->  -.  A  =  U. A )
2 ordeleqon 4596 . . . . . 6  |-  ( Ord 
A  <->  ( A  e.  On  \/  A  =  On ) )
3 unon 4638 . . . . . . . . 9  |-  U. On  =  On
43eqcomi 2300 . . . . . . . 8  |-  On  =  U. On
5 id 19 . . . . . . . 8  |-  ( A  =  On  ->  A  =  On )
6 unieq 3852 . . . . . . . 8  |-  ( A  =  On  ->  U. A  =  U. On )
74, 5, 63eqtr4a 2354 . . . . . . 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 4637 . . . 4  |-  ( Ord 
A  ->  ( A  =  U. A  \/  A  =  suc  U. A ) )
1312ord 366 . . 3  |-  ( Ord 
A  ->  ( -.  A  =  U. A  ->  A  =  suc  U. A
) )
14 onuni 4600 . . . 4  |-  ( A  e.  On  ->  U. A  e.  On )
15 ordtoplem.1 . . . 4  |-  ( U. A  e.  On  ->  suc  U. A  e.  S
)
16 eleq1a 2365 . . . 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 1632    e. wcel 1696    =/= wne 2459   U.cuni 3843   Ord word 4407   Oncon0 4408   suc csuc 4410
This theorem is referenced by:  ordtop  24947  ordtopcon  24950  ordtopt0  24953
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-suc 4414
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