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Theorem sspwuni 3867
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 2663 . . . 4  |-  x  e. 
_V
21elpw 3486 . . 3  |-  ( x  e.  ~P B  <->  x  C_  B
)
32ralbii 2418 . 2  |-  ( A. x  e.  A  x  e.  ~P B  <->  A. x  e.  A  x  C_  B
)
4 dfss3 3057 . 2  |-  ( A 
C_  ~P B  <->  A. x  e.  A  x  e.  ~P B )
5 unissb 3736 . 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 1465   A.wral 2393    C_ wss 3041   ~Pcpw 3480   U.cuni 3706
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-v 2662  df-in 3047  df-ss 3054  df-pw 3482  df-uni 3707
This theorem is referenced by:  pwssb  3868  elpwpw  3869  elpwuni  3872  rintm  3875  dftr4  4001  iotass  5075  tfrlemibfn  6193  tfr1onlembfn  6209  tfrcllembfn  6222  uniixp  6583  fipwssg  6835  unirnioo  9724  restid  12058  topgele  12123  topontopn  12131  unitg  12158  epttop  12186  resttopon  12267  txuni2  12352  txdis  12373  unirnblps  12518  unirnbl  12519
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