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Theorem iunpwss 4415
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 4367 . . 3  |-  ( E. x  e.  A  y 
C_  x  ->  y  C_ 
U_ x  e.  A  x )
2 eliun 4330 . . . 4  |-  ( y  e.  U_ x  e.  A  ~P x  <->  E. x  e.  A  y  e.  ~P x )
3 selpw 4017 . . . . 5  |-  ( y  e.  ~P x  <->  y  C_  x )
43rexbii 2965 . . . 4  |-  ( E. x  e.  A  y  e.  ~P x  <->  E. x  e.  A  y  C_  x )
52, 4bitri 249 . . 3  |-  ( y  e.  U_ x  e.  A  ~P x  <->  E. x  e.  A  y  C_  x )
6 selpw 4017 . . . 4  |-  ( y  e.  ~P U. A  <->  y 
C_  U. A )
7 uniiun 4378 . . . . 5  |-  U. A  =  U_ x  e.  A  x
87sseq2i 3529 . . . 4  |-  ( y 
C_  U. A  <->  y  C_  U_ x  e.  A  x )
96, 8bitri 249 . . 3  |-  ( y  e.  ~P U. A  <->  y 
C_  U_ x  e.  A  x )
101, 5, 93imtr4i 266 . 2  |-  ( y  e.  U_ x  e.  A  ~P x  -> 
y  e.  ~P U. A )
1110ssriv 3508 1  |-  U_ x  e.  A  ~P x  C_ 
~P U. A
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1767   E.wrex 2815    C_ wss 3476   ~Pcpw 4010   U.cuni 4245   U_ciun 4325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-v 3115  df-in 3483  df-ss 3490  df-pw 4012  df-uni 4246  df-iun 4327
This theorem is referenced by: (None)
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