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Theorem iunpwss 3993
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
Dummy variable  y is distinct from all other variables.

Proof of Theorem iunpwss
StepHypRef Expression
1 ssiun 3946 . . 3  |-  ( E. x  e.  A  y 
C_  x  ->  y  C_ 
U_ x  e.  A  x )
2 eliun 3911 . . . 4  |-  ( y  e.  U_ x  e.  A  ~P x  <->  E. x  e.  A  y  e.  ~P x )
3 vex 2793 . . . . . 6  |-  y  e. 
_V
43elpw 3633 . . . . 5  |-  ( y  e.  ~P x  <->  y  C_  x )
54rexbii 2570 . . . 4  |-  ( E. x  e.  A  y  e.  ~P x  <->  E. x  e.  A  y  C_  x )
62, 5bitri 242 . . 3  |-  ( y  e.  U_ x  e.  A  ~P x  <->  E. x  e.  A  y  C_  x )
73elpw 3633 . . . 4  |-  ( y  e.  ~P U. A  <->  y 
C_  U. A )
8 uniiun 3957 . . . . 5  |-  U. A  =  U_ x  e.  A  x
98sseq2i 3205 . . . 4  |-  ( y 
C_  U. A  <->  y  C_  U_ x  e.  A  x )
107, 9bitri 242 . . 3  |-  ( y  e.  ~P U. A  <->  y 
C_  U_ x  e.  A  x )
111, 6, 103imtr4i 259 . 2  |-  ( y  e.  U_ x  e.  A  ~P x  -> 
y  e.  ~P U. A )
1211ssriv 3186 1  |-  U_ x  e.  A  ~P x  C_ 
~P U. A
Colors of variables: wff set class
Syntax hints:    e. wcel 1685   E.wrex 2546    C_ wss 3154   ~Pcpw 3627   U.cuni 3829   U_ciun 3907
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ral 2550  df-rex 2551  df-v 2792  df-in 3161  df-ss 3168  df-pw 3629  df-uni 3830  df-iun 3909
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