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Theorem inttaror 26003
Description: The intersection of a Tarski's class with the class of the ordinal numbers is an ordinal number. (Contributed by FL, 20-Apr-2011.)
Assertion
Ref Expression
inttaror  |-  ( T  e.  Tarski  ->  ( On  i^i  T )  e.  On )

Proof of Theorem inttaror
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 incom 3374 . . 3  |-  ( T  i^i  On )  =  ( On  i^i  T
)
2 inex1g 4173 . . 3  |-  ( T  e.  Tarski  ->  ( T  i^i  On )  e.  _V )
31, 2syl5eqelr 2381 . 2  |-  ( T  e.  Tarski  ->  ( On  i^i  T )  e.  _V )
4 inss1 3402 . . . . . 6  |-  ( On 
i^i  T )  C_  On
54sseli 3189 . . . . 5  |-  ( x  e.  ( On  i^i  T )  ->  x  e.  On )
65adantl 452 . . . 4  |-  ( ( T  e.  Tarski  /\  x  e.  ( On  i^i  T
) )  ->  x  e.  On )
7 elin 3371 . . . . . 6  |-  ( x  e.  ( On  i^i  T )  <->  ( x  e.  On  /\  x  e.  T ) )
8 onss 4598 . . . . . . . 8  |-  ( x  e.  On  ->  x  C_  On )
9 eloni 4418 . . . . . . . . 9  |-  ( x  e.  On  ->  Ord  x )
10 ordtr 4422 . . . . . . . . 9  |-  ( Ord  x  ->  Tr  x
)
119, 10syl 15 . . . . . . . 8  |-  ( x  e.  On  ->  Tr  x )
12 tsktrss 8399 . . . . . . . . . . 11  |-  ( ( T  e.  Tarski  /\  Tr  x  /\  x  e.  T
)  ->  x  C_  T
)
13 ssin 3404 . . . . . . . . . . . . 13  |-  ( ( x  C_  On  /\  x  C_  T )  <->  x  C_  ( On  i^i  T ) )
1413biimpi 186 . . . . . . . . . . . 12  |-  ( ( x  C_  On  /\  x  C_  T )  ->  x  C_  ( On  i^i  T
) )
1514ex 423 . . . . . . . . . . 11  |-  ( x 
C_  On  ->  ( x 
C_  T  ->  x  C_  ( On  i^i  T
) ) )
1612, 15syl5com 26 . . . . . . . . . 10  |-  ( ( T  e.  Tarski  /\  Tr  x  /\  x  e.  T
)  ->  ( x  C_  On  ->  x  C_  ( On  i^i  T ) ) )
17163exp 1150 . . . . . . . . 9  |-  ( T  e.  Tarski  ->  ( Tr  x  ->  ( x  e.  T  ->  ( x  C_  On  ->  x  C_  ( On  i^i  T ) ) ) ) )
1817com14 82 . . . . . . . 8  |-  ( x 
C_  On  ->  ( Tr  x  ->  ( x  e.  T  ->  ( T  e.  Tarski  ->  x  C_  ( On  i^i  T ) ) ) ) )
198, 11, 18sylc 56 . . . . . . 7  |-  ( x  e.  On  ->  (
x  e.  T  -> 
( T  e.  Tarski  ->  x  C_  ( On  i^i  T ) ) ) )
2019imp 418 . . . . . 6  |-  ( ( x  e.  On  /\  x  e.  T )  ->  ( T  e.  Tarski  ->  x  C_  ( On  i^i  T ) ) )
217, 20sylbi 187 . . . . 5  |-  ( x  e.  ( On  i^i  T )  ->  ( T  e.  Tarski  ->  x  C_  ( On  i^i  T ) ) )
2221impcom 419 . . . 4  |-  ( ( T  e.  Tarski  /\  x  e.  ( On  i^i  T
) )  ->  x  C_  ( On  i^i  T
) )
236, 22jca 518 . . 3  |-  ( ( T  e.  Tarski  /\  x  e.  ( On  i^i  T
) )  ->  (
x  e.  On  /\  x  C_  ( On  i^i  T ) ) )
2423ralrimiva 2639 . 2  |-  ( T  e.  Tarski  ->  A. x  e.  ( On  i^i  T ) ( x  e.  On  /\  x  C_  ( On  i^i  T ) ) )
25 celsor 25214 . 2  |-  ( ( ( On  i^i  T
)  e.  _V  /\  A. x  e.  ( On 
i^i  T ) ( x  e.  On  /\  x  C_  ( On  i^i  T ) ) )  -> 
( On  i^i  T
)  e.  On )
263, 24, 25syl2anc 642 1  |-  ( T  e.  Tarski  ->  ( On  i^i  T )  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    e. wcel 1696   A.wral 2556   _Vcvv 2801    i^i cin 3164    C_ wss 3165   Tr wtr 4129   Ord word 4407   Oncon0 4408   Tarskictsk 8386
This theorem is referenced by:  inttarcar  26004  carinttar  26005  carinttar2  26006
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  ax-reg 7322
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-pw 3640  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-tsk 8387
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