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Theorem r1pwOLD 8276
Description: A stronger property of  R1 than rankpw 8273. The latter merely proves that  R1 of the successor is a power set, but here we prove that if  A is in the cumulative hierarchy, then  ~P A is in the cumulative hierarchy of the successor. (Contributed by Raph Levien, 29-May-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
r1pwOLD  |-  ( B  e.  On  ->  ( A  e.  ( R1 `  B )  <->  ~P A  e.  ( R1 `  suc  B ) ) )

Proof of Theorem r1pwOLD
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eleq1 2539 . . . . 5  |-  ( x  =  A  ->  (
x  e.  ( R1
`  B )  <->  A  e.  ( R1 `  B ) ) )
2 pweq 4019 . . . . . 6  |-  ( x  =  A  ->  ~P x  =  ~P A
)
32eleq1d 2536 . . . . 5  |-  ( x  =  A  ->  ( ~P x  e.  ( R1 `  suc  B )  <->  ~P A  e.  ( R1 `  suc  B ) ) )
41, 3bibi12d 321 . . . 4  |-  ( x  =  A  ->  (
( x  e.  ( R1 `  B )  <->  ~P x  e.  ( R1 `  suc  B ) )  <->  ( A  e.  ( R1 `  B
)  <->  ~P A  e.  ( R1 `  suc  B
) ) ) )
54imbi2d 316 . . 3  |-  ( x  =  A  ->  (
( B  e.  On  ->  ( x  e.  ( R1 `  B )  <->  ~P x  e.  ( R1 `  suc  B ) ) )  <->  ( B  e.  On  ->  ( A  e.  ( R1 `  B
)  <->  ~P A  e.  ( R1 `  suc  B
) ) ) ) )
6 vex 3121 . . . . . . 7  |-  x  e. 
_V
76rankr1a 8266 . . . . . 6  |-  ( B  e.  On  ->  (
x  e.  ( R1
`  B )  <->  ( rank `  x )  e.  B
) )
8 eloni 4894 . . . . . . 7  |-  ( B  e.  On  ->  Ord  B )
9 ordsucelsuc 6652 . . . . . . 7  |-  ( Ord 
B  ->  ( ( rank `  x )  e.  B  <->  suc  ( rank `  x
)  e.  suc  B
) )
108, 9syl 16 . . . . . 6  |-  ( B  e.  On  ->  (
( rank `  x )  e.  B  <->  suc  ( rank `  x
)  e.  suc  B
) )
117, 10bitrd 253 . . . . 5  |-  ( B  e.  On  ->  (
x  e.  ( R1
`  B )  <->  suc  ( rank `  x )  e.  suc  B ) )
126rankpw 8273 . . . . . 6  |-  ( rank `  ~P x )  =  suc  ( rank `  x
)
1312eleq1i 2544 . . . . 5  |-  ( (
rank `  ~P x
)  e.  suc  B  <->  suc  ( rank `  x
)  e.  suc  B
)
1411, 13syl6bbr 263 . . . 4  |-  ( B  e.  On  ->  (
x  e.  ( R1
`  B )  <->  ( rank `  ~P x )  e. 
suc  B ) )
15 suceloni 6643 . . . . 5  |-  ( B  e.  On  ->  suc  B  e.  On )
166pwex 4636 . . . . . 6  |-  ~P x  e.  _V
1716rankr1a 8266 . . . . 5  |-  ( suc 
B  e.  On  ->  ( ~P x  e.  ( R1 `  suc  B
)  <->  ( rank `  ~P x )  e.  suc  B ) )
1815, 17syl 16 . . . 4  |-  ( B  e.  On  ->  ( ~P x  e.  ( R1 `  suc  B )  <-> 
( rank `  ~P x
)  e.  suc  B
) )
1914, 18bitr4d 256 . . 3  |-  ( B  e.  On  ->  (
x  e.  ( R1
`  B )  <->  ~P x  e.  ( R1 `  suc  B ) ) )
205, 19vtoclg 3176 . 2  |-  ( A  e.  _V  ->  ( B  e.  On  ->  ( A  e.  ( R1
`  B )  <->  ~P A  e.  ( R1 `  suc  B ) ) ) )
21 elex 3127 . . . 4  |-  ( A  e.  ( R1 `  B )  ->  A  e.  _V )
22 elex 3127 . . . . 5  |-  ( ~P A  e.  ( R1
`  suc  B )  ->  ~P A  e.  _V )
23 pwexb 6606 . . . . 5  |-  ( A  e.  _V  <->  ~P A  e.  _V )
2422, 23sylibr 212 . . . 4  |-  ( ~P A  e.  ( R1
`  suc  B )  ->  A  e.  _V )
2521, 24pm5.21ni 352 . . 3  |-  ( -.  A  e.  _V  ->  ( A  e.  ( R1
`  B )  <->  ~P A  e.  ( R1 `  suc  B ) ) )
2625a1d 25 . 2  |-  ( -.  A  e.  _V  ->  ( B  e.  On  ->  ( A  e.  ( R1
`  B )  <->  ~P A  e.  ( R1 `  suc  B ) ) ) )
2720, 26pm2.61i 164 1  |-  ( B  e.  On  ->  ( A  e.  ( R1 `  B )  <->  ~P A  e.  ( R1 `  suc  B ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    = wceq 1379    e. wcel 1767   _Vcvv 3118   ~Pcpw 4016   Ord word 4883   Oncon0 4884   suc csuc 4886   ` cfv 5594   R1cr1 8192   rankcrnk 8193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-reg 8030  ax-inf2 8070
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6696  df-recs 7054  df-rdg 7088  df-r1 8194  df-rank 8195
This theorem is referenced by: (None)
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