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Theorem r1pwOLD 8262
Description: A stronger property of  R1 than rankpw 8259. 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 2513 . . . . 5  |-  ( x  =  A  ->  (
x  e.  ( R1
`  B )  <->  A  e.  ( R1 `  B ) ) )
2 pweq 3996 . . . . . 6  |-  ( x  =  A  ->  ~P x  =  ~P A
)
32eleq1d 2510 . . . . 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 3096 . . . . . . 7  |-  x  e. 
_V
76rankr1a 8252 . . . . . 6  |-  ( B  e.  On  ->  (
x  e.  ( R1
`  B )  <->  ( rank `  x )  e.  B
) )
8 eloni 4874 . . . . . . 7  |-  ( B  e.  On  ->  Ord  B )
9 ordsucelsuc 6638 . . . . . . 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 8259 . . . . . 6  |-  ( rank `  ~P x )  =  suc  ( rank `  x
)
1312eleq1i 2518 . . . . 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 6629 . . . . 5  |-  ( B  e.  On  ->  suc  B  e.  On )
166pwex 4616 . . . . . 6  |-  ~P x  e.  _V
1716rankr1a 8252 . . . . 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 3151 . 2  |-  ( A  e.  _V  ->  ( B  e.  On  ->  ( A  e.  ( R1
`  B )  <->  ~P A  e.  ( R1 `  suc  B ) ) ) )
21 elex 3102 . . . 4  |-  ( A  e.  ( R1 `  B )  ->  A  e.  _V )
22 elex 3102 . . . . 5  |-  ( ~P A  e.  ( R1
`  suc  B )  ->  ~P A  e.  _V )
23 pwexb 6592 . . . . 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 1381    e. wcel 1802   _Vcvv 3093   ~Pcpw 3993   Ord word 4863   Oncon0 4864   suc csuc 4866   ` cfv 5574   R1cr1 8178   rankcrnk 8179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-reg 8016  ax-inf2 8056
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-om 6682  df-recs 7040  df-rdg 7074  df-r1 8180  df-rank 8181
This theorem is referenced by: (None)
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