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

Theorem r1pwOLD 7608
Description: A stronger property of  R1 than rankpw 7605. 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 2418 . . . . 5  |-  ( x  =  A  ->  (
x  e.  ( R1
`  B )  <->  A  e.  ( R1 `  B ) ) )
2 pweq 3704 . . . . . 6  |-  ( x  =  A  ->  ~P x  =  ~P A
)
32eleq1d 2424 . . . . 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 2867 . . . . . . 7  |-  x  e. 
_V
76rankr1a 7598 . . . . . 6  |-  ( B  e.  On  ->  (
x  e.  ( R1
`  B )  <->  ( rank `  x )  e.  B
) )
8 eloni 4484 . . . . . . 7  |-  ( B  e.  On  ->  Ord  B )
9 ordsucelsuc 4695 . . . . . . 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 7605 . . . . . 6  |-  ( rank `  ~P x )  =  suc  ( rank `  x
)
1312eleq1i 2421 . . . . 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 4686 . . . . 5  |-  ( B  e.  On  ->  suc  B  e.  On )
166pwex 4274 . . . . . 6  |-  ~P x  e.  _V
1716rankr1a 7598 . . . . 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 2919 . 2  |-  ( A  e.  _V  ->  ( B  e.  On  ->  ( A  e.  ( R1
`  B )  <->  ~P A  e.  ( R1 `  suc  B ) ) ) )
21 elex 2872 . . . 4  |-  ( A  e.  ( R1 `  B )  ->  A  e.  _V )
22 elex 2872 . . . . 5  |-  ( ~P A  e.  ( R1
`  suc  B )  ->  ~P A  e.  _V )
23 pwexb 4646 . . . . 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 1642    e. wcel 1710   _Vcvv 2864   ~Pcpw 3701   Ord word 4473   Oncon0 4474   suc csuc 4476   ` cfv 5337   R1cr1 7524   rankcrnk 7525
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-reg 7396  ax-inf2 7432
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-recs 6475  df-rdg 6510  df-r1 7526  df-rank 7527
  Copyright terms: Public domain W3C validator