MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  r1pwOLD Unicode version

Theorem r1pwOLD 7518
Description: A stronger property of  R1 than rankpw 7515. 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 2343 . . . . 5  |-  ( x  =  A  ->  (
x  e.  ( R1
`  B )  <->  A  e.  ( R1 `  B ) ) )
2 pweq 3628 . . . . . 6  |-  ( x  =  A  ->  ~P x  =  ~P A
)
32eleq1d 2349 . . . . 5  |-  ( x  =  A  ->  ( ~P x  e.  ( R1 `  suc  B )  <->  ~P A  e.  ( R1 `  suc  B ) ) )
41, 3bibi12d 312 . . . 4  |-  ( x  =  A  ->  (
( x  e.  ( R1 `  B )  <->  ~P x  e.  ( R1 `  suc  B ) )  <->  ( A  e.  ( R1 `  B
)  <->  ~P A  e.  ( R1 `  suc  B
) ) ) )
54imbi2d 307 . . 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 2791 . . . . . . 7  |-  x  e. 
_V
76rankr1a 7508 . . . . . 6  |-  ( B  e.  On  ->  (
x  e.  ( R1
`  B )  <->  ( rank `  x )  e.  B
) )
8 eloni 4402 . . . . . . 7  |-  ( B  e.  On  ->  Ord  B )
9 ordsucelsuc 4613 . . . . . . 7  |-  ( Ord 
B  ->  ( ( rank `  x )  e.  B  <->  suc  ( rank `  x
)  e.  suc  B
) )
108, 9syl 15 . . . . . 6  |-  ( B  e.  On  ->  (
( rank `  x )  e.  B  <->  suc  ( rank `  x
)  e.  suc  B
) )
117, 10bitrd 244 . . . . 5  |-  ( B  e.  On  ->  (
x  e.  ( R1
`  B )  <->  suc  ( rank `  x )  e.  suc  B ) )
126rankpw 7515 . . . . . 6  |-  ( rank `  ~P x )  =  suc  ( rank `  x
)
1312eleq1i 2346 . . . . 5  |-  ( (
rank `  ~P x
)  e.  suc  B  <->  suc  ( rank `  x
)  e.  suc  B
)
1411, 13syl6bbr 254 . . . 4  |-  ( B  e.  On  ->  (
x  e.  ( R1
`  B )  <->  ( rank `  ~P x )  e. 
suc  B ) )
15 suceloni 4604 . . . . 5  |-  ( B  e.  On  ->  suc  B  e.  On )
166pwex 4193 . . . . . 6  |-  ~P x  e.  _V
1716rankr1a 7508 . . . . 5  |-  ( suc 
B  e.  On  ->  ( ~P x  e.  ( R1 `  suc  B
)  <->  ( rank `  ~P x )  e.  suc  B ) )
1815, 17syl 15 . . . 4  |-  ( B  e.  On  ->  ( ~P x  e.  ( R1 `  suc  B )  <-> 
( rank `  ~P x
)  e.  suc  B
) )
1914, 18bitr4d 247 . . 3  |-  ( B  e.  On  ->  (
x  e.  ( R1
`  B )  <->  ~P x  e.  ( R1 `  suc  B ) ) )
205, 19vtoclg 2843 . 2  |-  ( A  e.  _V  ->  ( B  e.  On  ->  ( A  e.  ( R1
`  B )  <->  ~P A  e.  ( R1 `  suc  B ) ) ) )
21 elex 2796 . . . 4  |-  ( A  e.  ( R1 `  B )  ->  A  e.  _V )
22 elex 2796 . . . . 5  |-  ( ~P A  e.  ( R1
`  suc  B )  ->  ~P A  e.  _V )
23 pwexb 4564 . . . . 5  |-  ( A  e.  _V  <->  ~P A  e.  _V )
2422, 23sylibr 203 . . . 4  |-  ( ~P A  e.  ( R1
`  suc  B )  ->  A  e.  _V )
2521, 24pm5.21ni 341 . . 3  |-  ( -.  A  e.  _V  ->  ( A  e.  ( R1
`  B )  <->  ~P A  e.  ( R1 `  suc  B ) ) )
2625a1d 22 . 2  |-  ( -.  A  e.  _V  ->  ( B  e.  On  ->  ( A  e.  ( R1
`  B )  <->  ~P A  e.  ( R1 `  suc  B ) ) ) )
2720, 26pm2.61i 156 1  |-  ( B  e.  On  ->  ( A  e.  ( R1 `  B )  <->  ~P A  e.  ( R1 `  suc  B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   _Vcvv 2788   ~Pcpw 3625   Ord word 4391   Oncon0 4392   suc csuc 4394   ` cfv 5255   R1cr1 7434   rankcrnk 7435
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-reg 7306  ax-inf2 7342
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-recs 6388  df-rdg 6423  df-r1 7436  df-rank 7437
  Copyright terms: Public domain W3C validator