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Theorem inttaror 25900
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 3361 . . 3  |-  ( T  i^i  On )  =  ( On  i^i  T
)
2 inex1g 4157 . . 3  |-  ( T  e.  Tarski  ->  ( T  i^i  On )  e.  _V )
31, 2syl5eqelr 2368 . 2  |-  ( T  e.  Tarski  ->  ( On  i^i  T )  e.  _V )
4 inss1 3389 . . . . . 6  |-  ( On 
i^i  T )  C_  On
54sseli 3176 . . . . 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 3358 . . . . . 6  |-  ( x  e.  ( On  i^i  T )  <->  ( x  e.  On  /\  x  e.  T ) )
8 onss 4582 . . . . . . . 8  |-  ( x  e.  On  ->  x  C_  On )
9 eloni 4402 . . . . . . . . 9  |-  ( x  e.  On  ->  Ord  x )
10 ordtr 4406 . . . . . . . . 9  |-  ( Ord  x  ->  Tr  x
)
119, 10syl 15 . . . . . . . 8  |-  ( x  e.  On  ->  Tr  x )
12 tsktrss 8383 . . . . . . . . . . 11  |-  ( ( T  e.  Tarski  /\  Tr  x  /\  x  e.  T
)  ->  x  C_  T
)
13 ssin 3391 . . . . . . . . . . . . 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 2626 . 2  |-  ( T  e.  Tarski  ->  A. x  e.  ( On  i^i  T ) ( x  e.  On  /\  x  C_  ( On  i^i  T ) ) )
25 celsor 25111 . 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 1684   A.wral 2543   _Vcvv 2788    i^i cin 3151    C_ wss 3152   Tr wtr 4113   Ord word 4391   Oncon0 4392   Tarskictsk 8370
This theorem is referenced by:  inttarcar  25901  carinttar  25902  carinttar2  25903
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  ax-reg 7306
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-pw 3627  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-tsk 8371
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