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Theorem r1pwOLD 8168
Description: A stronger property of  R1 than rankpw 8165. 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 2526 . . . . 5  |-  ( x  =  A  ->  (
x  e.  ( R1
`  B )  <->  A  e.  ( R1 `  B ) ) )
2 pweq 3974 . . . . . 6  |-  ( x  =  A  ->  ~P x  =  ~P A
)
32eleq1d 2523 . . . . 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 3081 . . . . . . 7  |-  x  e. 
_V
76rankr1a 8158 . . . . . 6  |-  ( B  e.  On  ->  (
x  e.  ( R1
`  B )  <->  ( rank `  x )  e.  B
) )
8 eloni 4840 . . . . . . 7  |-  ( B  e.  On  ->  Ord  B )
9 ordsucelsuc 6546 . . . . . . 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 8165 . . . . . 6  |-  ( rank `  ~P x )  =  suc  ( rank `  x
)
1312eleq1i 2531 . . . . 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 6537 . . . . 5  |-  ( B  e.  On  ->  suc  B  e.  On )
166pwex 4586 . . . . . 6  |-  ~P x  e.  _V
1716rankr1a 8158 . . . . 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 3136 . 2  |-  ( A  e.  _V  ->  ( B  e.  On  ->  ( A  e.  ( R1
`  B )  <->  ~P A  e.  ( R1 `  suc  B ) ) ) )
21 elex 3087 . . . 4  |-  ( A  e.  ( R1 `  B )  ->  A  e.  _V )
22 elex 3087 . . . . 5  |-  ( ~P A  e.  ( R1
`  suc  B )  ->  ~P A  e.  _V )
23 pwexb 6500 . . . . 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 1370    e. wcel 1758   _Vcvv 3078   ~Pcpw 3971   Ord word 4829   Oncon0 4830   suc csuc 4832   ` cfv 5529   R1cr1 8084   rankcrnk 8085
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-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-reg 7922  ax-inf2 7962
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-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-om 6590  df-recs 6945  df-rdg 6979  df-r1 8086  df-rank 8087
This theorem is referenced by: (None)
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