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Theorem sspwuni 3955
Description: Subclass relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
Assertion
Ref Expression
sspwuni  |-  ( A 
C_  ~P B  <->  U. A  C_  B )

Proof of Theorem sspwuni
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 vex 2733 . . . 4  |-  x  e. 
_V
21elpw 3570 . . 3  |-  ( x  e.  ~P B  <->  x  C_  B
)
32ralbii 2476 . 2  |-  ( A. x  e.  A  x  e.  ~P B  <->  A. x  e.  A  x  C_  B
)
4 dfss3 3137 . 2  |-  ( A 
C_  ~P B  <->  A. x  e.  A  x  e.  ~P B )
5 unissb 3824 . 2  |-  ( U. A  C_  B  <->  A. x  e.  A  x  C_  B
)
63, 4, 53bitr4i 211 1  |-  ( A 
C_  ~P B  <->  U. A  C_  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    e. wcel 2141   A.wral 2448    C_ wss 3121   ~Pcpw 3564   U.cuni 3794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-in 3127  df-ss 3134  df-pw 3566  df-uni 3795
This theorem is referenced by:  pwssb  3956  elpwpw  3957  elpwuni  3960  rintm  3963  dftr4  4090  iotass  5175  tfrlemibfn  6304  tfr1onlembfn  6320  tfrcllembfn  6333  uniixp  6695  fipwssg  6952  unirnioo  9917  restid  12577  topgele  12780  topontopn  12788  unitg  12815  epttop  12843  resttopon  12924  txuni2  13009  txdis  13030  unirnblps  13175  unirnbl  13176
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