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Theorem psshepw 36455
Description: The relation between sets and their proper subsets is hereditary in the powerclass of any class. (Contributed by RP, 28-Mar-2020.)
Assertion
Ref Expression
psshepw  |-  `' [ C.] hereditary  ~P A

Proof of Theorem psshepw
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfhe3 36441 . 2  |-  ( `' [ C.] hereditary 
~P A  <->  A. x
( x  e.  ~P A  ->  A. y ( x `' [ C.]  y  ->  y  e.  ~P A ) ) )
2 pssss 3514 . . . . . 6  |-  ( y 
C.  x  ->  y  C_  x )
3 sstr2 3425 . . . . . 6  |-  ( y 
C_  x  ->  (
x  C_  A  ->  y 
C_  A ) )
42, 3syl 17 . . . . 5  |-  ( y 
C.  x  ->  (
x  C_  A  ->  y 
C_  A ) )
54com12 31 . . . 4  |-  ( x 
C_  A  ->  (
y  C.  x  ->  y 
C_  A ) )
65alrimiv 1781 . . 3  |-  ( x 
C_  A  ->  A. y
( y  C.  x  ->  y  C_  A )
)
7 selpw 3949 . . 3  |-  ( x  e.  ~P A  <->  x  C_  A
)
8 vex 3034 . . . . . . 7  |-  x  e. 
_V
9 vex 3034 . . . . . . 7  |-  y  e. 
_V
108, 9brcnv 5022 . . . . . 6  |-  ( x `' [ C.]  y  <->  y [ C.]  x
)
118brrpss 6593 . . . . . 6  |-  ( y [ C.]  x  <->  y  C.  x
)
1210, 11bitri 257 . . . . 5  |-  ( x `' [ C.]  y  <->  y  C.  x
)
13 selpw 3949 . . . . 5  |-  ( y  e.  ~P A  <->  y  C_  A )
1412, 13imbi12i 333 . . . 4  |-  ( ( x `' [ C.]  y  ->  y  e.  ~P A
)  <->  ( y  C.  x  ->  y  C_  A
) )
1514albii 1699 . . 3  |-  ( A. y ( x `' [ C.]  y  ->  y  e. 
~P A )  <->  A. y
( y  C.  x  ->  y  C_  A )
)
166, 7, 153imtr4i 274 . 2  |-  ( x  e.  ~P A  ->  A. y ( x `' [ C.]  y  ->  y  e. 
~P A ) )
171, 16mpgbir 1681 1  |-  `' [ C.] hereditary  ~P A
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1450    e. wcel 1904    C_ wss 3390    C. wpss 3391   ~Pcpw 3942   class class class wbr 4395   `'ccnv 4838   [ C.] crpss 6589   hereditary whe 36438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-xp 4845  df-rel 4846  df-cnv 4847  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-rpss 6590  df-he 36439
This theorem is referenced by:  sshepw  36456
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