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Theorem tsksuc 9129
Description: If an element of a Tarski 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 994 . 2  |-  ( ( T  e.  Tarski  /\  A  e.  On  /\  A  e.  T )  ->  T  e.  Tarski )
2 tskpw 9120 . . 3  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ~P A  e.  T )
323adant2 1013 . 2  |-  ( ( T  e.  Tarski  /\  A  e.  On  /\  A  e.  T )  ->  ~P A  e.  T )
4 eloni 4877 . . . . 5  |-  ( A  e.  On  ->  Ord  A )
543ad2ant2 1016 . . . 4  |-  ( ( T  e.  Tarski  /\  A  e.  On  /\  A  e.  T )  ->  Ord  A )
6 ordunisuc 6640 . . . 4  |-  ( Ord 
A  ->  U. suc  A  =  A )
7 eqimss 3541 . . . 4  |-  ( U. suc  A  =  A  ->  U. suc  A  C_  A
)
85, 6, 73syl 20 . . 3  |-  ( ( T  e.  Tarski  /\  A  e.  On  /\  A  e.  T )  ->  U. suc  A 
C_  A )
9 sspwuni 4404 . . 3  |-  ( suc 
A  C_  ~P A  <->  U.
suc  A  C_  A )
108, 9sylibr 212 . 2  |-  ( ( T  e.  Tarski  /\  A  e.  On  /\  A  e.  T )  ->  suc  A 
C_  ~P A )
11 tskss 9125 . 2  |-  ( ( T  e.  Tarski  /\  ~P A  e.  T  /\  suc  A  C_  ~P A
)  ->  suc  A  e.  T )
121, 3, 10, 11syl3anc 1226 1  |-  ( ( T  e.  Tarski  /\  A  e.  On  /\  A  e.  T )  ->  suc  A  e.  T )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 971    = wceq 1398    e. wcel 1823    C_ wss 3461   ~Pcpw 3999   U.cuni 4235   Ord word 4866   Oncon0 4867   suc csuc 4869   Tarskictsk 9115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-tr 4533  df-eprel 4780  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-suc 4873  df-tsk 9116
This theorem is referenced by:  tsk2  9132
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