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Theorem ordtoplem 31044
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 2601 . 2  |-  ( A  =/=  U. A  <->  -.  A  =  U. A )
2 ordeleqon 6573 . . . . . 6  |-  ( Ord 
A  <->  ( A  e.  On  \/  A  =  On ) )
3 unon 6616 . . . . . . . . 9  |-  U. On  =  On
43eqcomi 2437 . . . . . . . 8  |-  On  =  U. On
5 id 22 . . . . . . . 8  |-  ( A  =  On  ->  A  =  On )
6 unieq 4170 . . . . . . . 8  |-  ( A  =  On  ->  U. A  =  U. On )
74, 5, 63eqtr4a 2488 . . . . . . 7  |-  ( A  =  On  ->  A  =  U. A )
87orim2i 520 . . . . . 6  |-  ( ( A  e.  On  \/  A  =  On )  ->  ( A  e.  On  \/  A  =  U. A ) )
92, 8sylbi 198 . . . . 5  |-  ( Ord 
A  ->  ( A  e.  On  \/  A  = 
U. A ) )
109orcomd 389 . . . 4  |-  ( Ord 
A  ->  ( A  =  U. A  \/  A  e.  On ) )
1110ord 378 . . 3  |-  ( Ord 
A  ->  ( -.  A  =  U. A  ->  A  e.  On )
)
12 orduniorsuc 6615 . . . 4  |-  ( Ord 
A  ->  ( A  =  U. A  \/  A  =  suc  U. A ) )
1312ord 378 . . 3  |-  ( Ord 
A  ->  ( -.  A  =  U. A  ->  A  =  suc  U. A
) )
14 onuni 6578 . . . 4  |-  ( A  e.  On  ->  U. A  e.  On )
15 ordtoplem.1 . . . 4  |-  ( U. A  e.  On  ->  suc  U. A  e.  S
)
16 eleq1a 2501 . . . 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 66 . 2  |-  ( Ord 
A  ->  ( -.  A  =  U. A  ->  A  e.  S )
)
191, 18syl5bi 220 1  |-  ( Ord 
A  ->  ( A  =/=  U. A  ->  A  e.  S ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 369    = wceq 1437    e. wcel 1872    =/= wne 2599   U.cuni 4162   Ord word 5384   Oncon0 5385   suc csuc 5387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pr 4603  ax-un 6541
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-tr 4462  df-eprel 4707  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-ord 5388  df-on 5389  df-suc 5391
This theorem is referenced by:  ordtop  31045  ordtopcon  31048  ordtopt0  31051
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