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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

Proof of Theorem iunpwss
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ssiun 3943 . . 3  |-  ( E. x  e.  A  y 
C_  x  ->  y  C_ 
U_ x  e.  A  x )
2 eliun 3905 . . . 4  |-  ( y  e.  U_ x  e.  A  ~P x  <->  E. x  e.  A  y  e.  ~P x )
3 vex 2755 . . . . . 6  |-  y  e. 
_V
43elpw 3596 . . . . 5  |-  ( y  e.  ~P x  <->  y  C_  x )
54rexbii 2497 . . . 4  |-  ( E. x  e.  A  y  e.  ~P x  <->  E. x  e.  A  y  C_  x )
62, 5bitri 184 . . 3  |-  ( y  e.  U_ x  e.  A  ~P x  <->  E. x  e.  A  y  C_  x )
73elpw 3596 . . . 4  |-  ( y  e.  ~P U. A  <->  y 
C_  U. A )
8 uniiun 3955 . . . . 5  |-  U. A  =  U_ x  e.  A  x
98sseq2i 3197 . . . 4  |-  ( y 
C_  U. A  <->  y  C_  U_ x  e.  A  x )
107, 9bitri 184 . . 3  |-  ( y  e.  ~P U. A  <->  y 
C_  U_ x  e.  A  x )
111, 6, 103imtr4i 201 . 2  |-  ( y  e.  U_ x  e.  A  ~P x  -> 
y  e.  ~P U. A )
1211ssriv 3174 1  |-  U_ x  e.  A  ~P x  C_ 
~P U. A
Colors of variables: wff set class
Syntax hints:    e. wcel 2160   E.wrex 2469    C_ wss 3144   ~Pcpw 3590   U.cuni 3824   U_ciun 3901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-in 3150  df-ss 3157  df-pw 3592  df-uni 3825  df-iun 3903
This theorem is referenced by: (None)
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