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Theorem iunpwss 4140
Description: Inclusion of an indexed union of a power class in the power class of the union of its index. Part of Exercise 24(b) of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.)
Assertion
Ref Expression
iunpwss  |-  U_ x  e.  A  ~P x  C_ 
~P U. A
Distinct variable group:    x, A

Proof of Theorem iunpwss
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ssiun 4093 . . 3  |-  ( E. x  e.  A  y 
C_  x  ->  y  C_ 
U_ x  e.  A  x )
2 eliun 4057 . . . 4  |-  ( y  e.  U_ x  e.  A  ~P x  <->  E. x  e.  A  y  e.  ~P x )
3 vex 2919 . . . . . 6  |-  y  e. 
_V
43elpw 3765 . . . . 5  |-  ( y  e.  ~P x  <->  y  C_  x )
54rexbii 2691 . . . 4  |-  ( E. x  e.  A  y  e.  ~P x  <->  E. x  e.  A  y  C_  x )
62, 5bitri 241 . . 3  |-  ( y  e.  U_ x  e.  A  ~P x  <->  E. x  e.  A  y  C_  x )
73elpw 3765 . . . 4  |-  ( y  e.  ~P U. A  <->  y 
C_  U. A )
8 uniiun 4104 . . . . 5  |-  U. A  =  U_ x  e.  A  x
98sseq2i 3333 . . . 4  |-  ( y 
C_  U. A  <->  y  C_  U_ x  e.  A  x )
107, 9bitri 241 . . 3  |-  ( y  e.  ~P U. A  <->  y 
C_  U_ x  e.  A  x )
111, 6, 103imtr4i 258 . 2  |-  ( y  e.  U_ x  e.  A  ~P x  -> 
y  e.  ~P U. A )
1211ssriv 3312 1  |-  U_ x  e.  A  ~P x  C_ 
~P U. A
Colors of variables: wff set class
Syntax hints:    e. wcel 1721   E.wrex 2667    C_ wss 3280   ~Pcpw 3759   U.cuni 3975   U_ciun 4053
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rex 2672  df-v 2918  df-in 3287  df-ss 3294  df-pw 3761  df-uni 3976  df-iun 4055
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