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Theorem inttarcar 26004
Description: The intersection of a Tarski's class and the ordinal numbers is equipotent to the Tarski's class. JFM CLASSES2. . (Contributed by FL, 20-Apr-2011.)
Assertion
Ref Expression
inttarcar  |-  ( T  e.  Tarski  ->  ( On  i^i  T )  ~~  T )

Proof of Theorem inttarcar
StepHypRef Expression
1 inss2 3403 . 2  |-  ( On 
i^i  T )  C_  T
2 tsken 8392 . . 3  |-  ( ( T  e.  Tarski  /\  ( On  i^i  T )  C_  T )  ->  (
( On  i^i  T
)  ~~  T  \/  ( On  i^i  T )  e.  T ) )
3 ax-1 5 . . . 4  |-  ( ( On  i^i  T ) 
~~  T  ->  (
( T  e.  Tarski  /\  ( On  i^i  T
)  C_  T )  ->  ( On  i^i  T
)  ~~  T )
)
4 inttaror 26003 . . . . . . 7  |-  ( T  e.  Tarski  ->  ( On  i^i  T )  e.  On )
5 elin 3371 . . . . . . . . 9  |-  ( ( On  i^i  T )  e.  ( On  i^i  T )  <->  ( ( On 
i^i  T )  e.  On  /\  ( On 
i^i  T )  e.  T ) )
6 elirr 7328 . . . . . . . . . 10  |-  -.  ( On  i^i  T )  e.  ( On  i^i  T
)
76pm2.21i 123 . . . . . . . . 9  |-  ( ( On  i^i  T )  e.  ( On  i^i  T )  ->  ( On  i^i  T )  ~~  T
)
85, 7sylbir 204 . . . . . . . 8  |-  ( ( ( On  i^i  T
)  e.  On  /\  ( On  i^i  T )  e.  T )  -> 
( On  i^i  T
)  ~~  T )
98ex 423 . . . . . . 7  |-  ( ( On  i^i  T )  e.  On  ->  (
( On  i^i  T
)  e.  T  -> 
( On  i^i  T
)  ~~  T )
)
104, 9syl 15 . . . . . 6  |-  ( T  e.  Tarski  ->  ( ( On 
i^i  T )  e.  T  ->  ( On  i^i  T )  ~~  T
) )
1110adantr 451 . . . . 5  |-  ( ( T  e.  Tarski  /\  ( On  i^i  T )  C_  T )  ->  (
( On  i^i  T
)  e.  T  -> 
( On  i^i  T
)  ~~  T )
)
1211com12 27 . . . 4  |-  ( ( On  i^i  T )  e.  T  ->  (
( T  e.  Tarski  /\  ( On  i^i  T
)  C_  T )  ->  ( On  i^i  T
)  ~~  T )
)
133, 12jaoi 368 . . 3  |-  ( ( ( On  i^i  T
)  ~~  T  \/  ( On  i^i  T )  e.  T )  -> 
( ( T  e. 
Tarski  /\  ( On  i^i  T )  C_  T )  ->  ( On  i^i  T
)  ~~  T )
)
142, 13mpcom 32 . 2  |-  ( ( T  e.  Tarski  /\  ( On  i^i  T )  C_  T )  ->  ( On  i^i  T )  ~~  T )
151, 14mpan2 652 1  |-  ( T  e.  Tarski  ->  ( On  i^i  T )  ~~  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    e. wcel 1696    i^i cin 3164    C_ wss 3165   class class class wbr 4039   Oncon0 4408    ~~ cen 6876   Tarskictsk 8386
This theorem is referenced by:  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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