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Theorem tsksuc 8400
Description: If an element of a Tarski's class is an ordinal number, its successor is an element of the class. JFM CLASSES2 th. 6 (partly). (Contributed by FL, 22-Feb-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
Assertion
Ref Expression
tsksuc  |-  ( ( T  e.  Tarski  /\  A  e.  On  /\  A  e.  T )  ->  suc  A  e.  T )

Proof of Theorem tsksuc
StepHypRef Expression
1 simp1 955 . 2  |-  ( ( T  e.  Tarski  /\  A  e.  On  /\  A  e.  T )  ->  T  e.  Tarski )
2 tskpw 8391 . . 3  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ~P A  e.  T )
323adant2 974 . 2  |-  ( ( T  e.  Tarski  /\  A  e.  On  /\  A  e.  T )  ->  ~P A  e.  T )
4 eloni 4418 . . . . 5  |-  ( A  e.  On  ->  Ord  A )
543ad2ant2 977 . . . 4  |-  ( ( T  e.  Tarski  /\  A  e.  On  /\  A  e.  T )  ->  Ord  A )
6 ordunisuc 4639 . . . 4  |-  ( Ord 
A  ->  U. suc  A  =  A )
7 eqimss 3243 . . . 4  |-  ( U. suc  A  =  A  ->  U. suc  A  C_  A
)
85, 6, 73syl 18 . . 3  |-  ( ( T  e.  Tarski  /\  A  e.  On  /\  A  e.  T )  ->  U. suc  A 
C_  A )
9 sspwuni 4003 . . 3  |-  ( suc 
A  C_  ~P A  <->  U.
suc  A  C_  A )
108, 9sylibr 203 . 2  |-  ( ( T  e.  Tarski  /\  A  e.  On  /\  A  e.  T )  ->  suc  A 
C_  ~P A )
11 tskss 8396 . 2  |-  ( ( T  e.  Tarski  /\  ~P A  e.  T  /\  suc  A  C_  ~P A
)  ->  suc  A  e.  T )
121, 3, 10, 11syl3anc 1182 1  |-  ( ( T  e.  Tarski  /\  A  e.  On  /\  A  e.  T )  ->  suc  A  e.  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1632    e. wcel 1696    C_ wss 3165   ~Pcpw 3638   U.cuni 3843   Ord word 4407   Oncon0 4408   suc csuc 4410   Tarskictsk 8386
This theorem is referenced by:  tsk2  8403
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-pow 4204  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-suc 4414  df-tsk 8387
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