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Theorem sspwuni 3768
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 2605 . . . 4  |-  x  e. 
_V
21elpw 3396 . . 3  |-  ( x  e.  ~P B  <->  x  C_  B
)
32ralbii 2373 . 2  |-  ( A. x  e.  A  x  e.  ~P B  <->  A. x  e.  A  x  C_  B
)
4 dfss3 2990 . 2  |-  ( A 
C_  ~P B  <->  A. x  e.  A  x  e.  ~P B )
5 unissb 3639 . 2  |-  ( U. A  C_  B  <->  A. x  e.  A  x  C_  B
)
63, 4, 53bitr4i 210 1  |-  ( A 
C_  ~P B  <->  U. A  C_  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 103    e. wcel 1434   A.wral 2349    C_ wss 2974   ~Pcpw 3390   U.cuni 3609
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-v 2604  df-in 2980  df-ss 2987  df-pw 3392  df-uni 3610
This theorem is referenced by:  pwssb  3769  elpwuni  3770  rintm  3773  dftr4  3888  iotass  4914  tfrlemibfn  5977  tfr1onlembfn  5993  tfrcllembfn  6006  unirnioo  9072
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