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Theorem partarelt2 25897
Description: If  Z is an element of our tar function at  A then  ~P Z is an element or tar at  suc  A. CLASSES1 th. 15 (Contributed by FL, 13-Apr-2011.)
Assertion
Ref Expression
partarelt2  |-  ( ( X  e.  B  /\  Y  e.  On  /\  suc  A  e.  Y )  -> 
( Z  e.  ( ( tar `  <. X ,  Y >. ) `  A )  ->  ~P Z  e.  ( ( tar `  <. X ,  Y >. ) `  suc  A
) ) )

Proof of Theorem partarelt2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqidd 2284 . . . . . 6  |-  ( ( X  e.  B  /\  Y  e.  On  /\  suc  A  e.  Y )  ->  ~P Z  =  ~P Z )
21olcd 382 . . . . 5  |-  ( ( X  e.  B  /\  Y  e.  On  /\  suc  A  e.  Y )  -> 
( ~P Z  C_  Z  \/  ~P Z  =  ~P Z ) )
3 sseq2 3200 . . . . . . 7  |-  ( x  =  Z  ->  ( ~P Z  C_  x  <->  ~P Z  C_  Z ) )
4 pweq 3628 . . . . . . . 8  |-  ( x  =  Z  ->  ~P x  =  ~P Z
)
54eqeq2d 2294 . . . . . . 7  |-  ( x  =  Z  ->  ( ~P Z  =  ~P x 
<->  ~P Z  =  ~P Z ) )
63, 5orbi12d 690 . . . . . 6  |-  ( x  =  Z  ->  (
( ~P Z  C_  x  \/  ~P Z  =  ~P x )  <->  ( ~P Z  C_  Z  \/  ~P Z  =  ~P Z
) ) )
76rspcev 2884 . . . . 5  |-  ( ( Z  e.  ( ( tar `  <. X ,  Y >. ) `  A
)  /\  ( ~P Z  C_  Z  \/  ~P Z  =  ~P Z
) )  ->  E. x  e.  ( ( tar `  <. X ,  Y >. ) `  A ) ( ~P Z  C_  x  \/  ~P Z  =  ~P x ) )
82, 7sylan2 460 . . . 4  |-  ( ( Z  e.  ( ( tar `  <. X ,  Y >. ) `  A
)  /\  ( X  e.  B  /\  Y  e.  On  /\  suc  A  e.  Y ) )  ->  E. x  e.  (
( tar `  <. X ,  Y >. ) `  A ) ( ~P Z  C_  x  \/  ~P Z  =  ~P x ) )
98olcd 382 . . 3  |-  ( ( Z  e.  ( ( tar `  <. X ,  Y >. ) `  A
)  /\  ( X  e.  B  /\  Y  e.  On  /\  suc  A  e.  Y ) )  -> 
( ( ~P Z  C_  ( ( tar `  <. X ,  Y >. ) `  A )  /\  ~P Z  e.  ( tarskiMap `  X
) )  \/  E. x  e.  ( ( tar `  <. X ,  Y >. ) `  A ) ( ~P Z  C_  x  \/  ~P Z  =  ~P x ) ) )
109expcom 424 . 2  |-  ( ( X  e.  B  /\  Y  e.  On  /\  suc  A  e.  Y )  -> 
( Z  e.  ( ( tar `  <. X ,  Y >. ) `  A )  ->  (
( ~P Z  C_  ( ( tar `  <. X ,  Y >. ) `  A )  /\  ~P Z  e.  ( tarskiMap `  X
) )  \/  E. x  e.  ( ( tar `  <. X ,  Y >. ) `  A ) ( ~P Z  C_  x  \/  ~P Z  =  ~P x ) ) ) )
11 vtarsuelt 25895 . 2  |-  ( ( X  e.  B  /\  Y  e.  On  /\  suc  A  e.  Y )  -> 
( ~P Z  e.  ( ( tar `  <. X ,  Y >. ) `  suc  A )  <->  ( ( ~P Z  C_  ( ( tar `  <. X ,  Y >. ) `  A
)  /\  ~P Z  e.  ( tarskiMap `  X )
)  \/  E. x  e.  ( ( tar `  <. X ,  Y >. ) `  A ) ( ~P Z  C_  x  \/  ~P Z  =  ~P x ) ) ) )
1210, 11sylibrd 225 1  |-  ( ( X  e.  B  /\  Y  e.  On  /\  suc  A  e.  Y )  -> 
( Z  e.  ( ( tar `  <. X ,  Y >. ) `  A )  ->  ~P Z  e.  ( ( tar `  <. X ,  Y >. ) `  suc  A
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544    C_ wss 3152   ~Pcpw 3625   <.cop 3643   Oncon0 4392   suc csuc 4394   ` cfv 5255   tarskiMapctskm 8459   tarctar 25881
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-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-groth 8445
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-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  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-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-tsk 8371  df-tskm 8460  df-tar 25882
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