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Theorem eltintpar 26002
Description: An element of the intersection of a Tarski's class with the class of the ordinal numbers is a part of the intersection. (Contributed by FL, 20-Apr-2011.)
Assertion
Ref Expression
eltintpar  |-  ( T  e.  Tarski  ->  ( A  e.  ( On  i^i  T
)  ->  A  C_  ( On  i^i  T ) ) )

Proof of Theorem eltintpar
StepHypRef Expression
1 elin 3371 . . . . 5  |-  ( A  e.  ( On  i^i  T )  <->  ( A  e.  On  /\  A  e.  T ) )
2 onss 4598 . . . . . . . 8  |-  ( A  e.  On  ->  A  C_  On )
32ad2antrr 706 . . . . . . 7  |-  ( ( ( A  e.  On  /\  A  e.  T )  /\  T  e.  Tarski )  ->  A  C_  On )
4 eloni 4418 . . . . . . . . 9  |-  ( A  e.  On  ->  Ord  A )
5 ordtr 4422 . . . . . . . . 9  |-  ( Ord 
A  ->  Tr  A
)
6 tsktrss 8399 . . . . . . . . . . 11  |-  ( ( T  e.  Tarski  /\  Tr  A  /\  A  e.  T
)  ->  A  C_  T
)
763exp 1150 . . . . . . . . . 10  |-  ( T  e.  Tarski  ->  ( Tr  A  ->  ( A  e.  T  ->  A  C_  T )
) )
87com3l 75 . . . . . . . . 9  |-  ( Tr  A  ->  ( A  e.  T  ->  ( T  e.  Tarski  ->  A  C_  T
) ) )
94, 5, 83syl 18 . . . . . . . 8  |-  ( A  e.  On  ->  ( A  e.  T  ->  ( T  e.  Tarski  ->  A  C_  T ) ) )
109imp31 421 . . . . . . 7  |-  ( ( ( A  e.  On  /\  A  e.  T )  /\  T  e.  Tarski )  ->  A  C_  T
)
113, 10jca 518 . . . . . 6  |-  ( ( ( A  e.  On  /\  A  e.  T )  /\  T  e.  Tarski )  ->  ( A  C_  On  /\  A  C_  T
) )
1211ex 423 . . . . 5  |-  ( ( A  e.  On  /\  A  e.  T )  ->  ( T  e.  Tarski  -> 
( A  C_  On  /\  A  C_  T )
) )
131, 12sylbi 187 . . . 4  |-  ( A  e.  ( On  i^i  T )  ->  ( T  e.  Tarski  ->  ( A  C_  On  /\  A  C_  T
) ) )
1413impcom 419 . . 3  |-  ( ( T  e.  Tarski  /\  A  e.  ( On  i^i  T
) )  ->  ( A  C_  On  /\  A  C_  T ) )
15 ssin 3404 . . 3  |-  ( ( A  C_  On  /\  A  C_  T )  <->  A  C_  ( On  i^i  T ) )
1614, 15sylib 188 . 2  |-  ( ( T  e.  Tarski  /\  A  e.  ( On  i^i  T
) )  ->  A  C_  ( On  i^i  T
) )
1716ex 423 1  |-  ( T  e.  Tarski  ->  ( A  e.  ( On  i^i  T
)  ->  A  C_  ( On  i^i  T ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1696    i^i cin 3164    C_ wss 3165   Tr wtr 4129   Ord word 4407   Oncon0 4408   Tarskictsk 8386
This theorem is referenced by:  carinttar  26005
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-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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