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Theorem tsksuc 8384
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 8375 . . 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 4402 . . . . 5  |-  ( A  e.  On  ->  Ord  A )
543ad2ant2 977 . . . 4  |-  ( ( T  e.  Tarski  /\  A  e.  On  /\  A  e.  T )  ->  Ord  A )
6 ordunisuc 4623 . . . 4  |-  ( Ord 
A  ->  U. suc  A  =  A )
7 eqimss 3230 . . . 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 3987 . . 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 8380 . 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 1623    e. wcel 1684    C_ wss 3152   ~Pcpw 3625   U.cuni 3827   Ord word 4391   Oncon0 4392   suc csuc 4394   Tarskictsk 8370
This theorem is referenced by:  tsk2  8387
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398  df-tsk 8371
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