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Theorem tsksuc 8637
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 957 . 2  |-  ( ( T  e.  Tarski  /\  A  e.  On  /\  A  e.  T )  ->  T  e.  Tarski )
2 tskpw 8628 . . 3  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ~P A  e.  T )
323adant2 976 . 2  |-  ( ( T  e.  Tarski  /\  A  e.  On  /\  A  e.  T )  ->  ~P A  e.  T )
4 eloni 4591 . . . . 5  |-  ( A  e.  On  ->  Ord  A )
543ad2ant2 979 . . . 4  |-  ( ( T  e.  Tarski  /\  A  e.  On  /\  A  e.  T )  ->  Ord  A )
6 ordunisuc 4812 . . . 4  |-  ( Ord 
A  ->  U. suc  A  =  A )
7 eqimss 3400 . . . 4  |-  ( U. suc  A  =  A  ->  U. suc  A  C_  A
)
85, 6, 73syl 19 . . 3  |-  ( ( T  e.  Tarski  /\  A  e.  On  /\  A  e.  T )  ->  U. suc  A 
C_  A )
9 sspwuni 4176 . . 3  |-  ( suc 
A  C_  ~P A  <->  U.
suc  A  C_  A )
108, 9sylibr 204 . 2  |-  ( ( T  e.  Tarski  /\  A  e.  On  /\  A  e.  T )  ->  suc  A 
C_  ~P A )
11 tskss 8633 . 2  |-  ( ( T  e.  Tarski  /\  ~P A  e.  T  /\  suc  A  C_  ~P A
)  ->  suc  A  e.  T )
121, 3, 10, 11syl3anc 1184 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 936    = wceq 1652    e. wcel 1725    C_ wss 3320   ~Pcpw 3799   U.cuni 4015   Ord word 4580   Oncon0 4581   suc csuc 4583   Tarskictsk 8623
This theorem is referenced by:  tsk2  8640
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-tr 4303  df-eprel 4494  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-suc 4587  df-tsk 8624
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