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Theorem tsksuc 9041
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 988 . 2  |-  ( ( T  e.  Tarski  /\  A  e.  On  /\  A  e.  T )  ->  T  e.  Tarski )
2 tskpw 9032 . . 3  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ~P A  e.  T )
323adant2 1007 . 2  |-  ( ( T  e.  Tarski  /\  A  e.  On  /\  A  e.  T )  ->  ~P A  e.  T )
4 eloni 4838 . . . . 5  |-  ( A  e.  On  ->  Ord  A )
543ad2ant2 1010 . . . 4  |-  ( ( T  e.  Tarski  /\  A  e.  On  /\  A  e.  T )  ->  Ord  A )
6 ordunisuc 6554 . . . 4  |-  ( Ord 
A  ->  U. suc  A  =  A )
7 eqimss 3517 . . . 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 4365 . . 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 9037 . 2  |-  ( ( T  e.  Tarski  /\  ~P A  e.  T  /\  suc  A  C_  ~P A
)  ->  suc  A  e.  T )
121, 3, 10, 11syl3anc 1219 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 965    = wceq 1370    e. wcel 1758    C_ wss 3437   ~Pcpw 3969   U.cuni 4200   Ord word 4827   Oncon0 4828   suc csuc 4830   Tarskictsk 9027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-tr 4495  df-eprel 4741  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-suc 4834  df-tsk 9028
This theorem is referenced by:  tsk2  9044
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