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Theorem iunpwss 3826
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 3778 . . 3  |-  ( E. x  e.  A  y 
C_  x  ->  y  C_ 
U_ x  e.  A  x )
2 eliun 3740 . . . 4  |-  ( y  e.  U_ x  e.  A  ~P x  <->  E. x  e.  A  y  e.  ~P x )
3 vex 2623 . . . . . 6  |-  y  e. 
_V
43elpw 3439 . . . . 5  |-  ( y  e.  ~P x  <->  y  C_  x )
54rexbii 2386 . . . 4  |-  ( E. x  e.  A  y  e.  ~P x  <->  E. x  e.  A  y  C_  x )
62, 5bitri 183 . . 3  |-  ( y  e.  U_ x  e.  A  ~P x  <->  E. x  e.  A  y  C_  x )
73elpw 3439 . . . 4  |-  ( y  e.  ~P U. A  <->  y 
C_  U. A )
8 uniiun 3789 . . . . 5  |-  U. A  =  U_ x  e.  A  x
98sseq2i 3052 . . . 4  |-  ( y 
C_  U. A  <->  y  C_  U_ x  e.  A  x )
107, 9bitri 183 . . 3  |-  ( y  e.  ~P U. A  <->  y 
C_  U_ x  e.  A  x )
111, 6, 103imtr4i 200 . 2  |-  ( y  e.  U_ x  e.  A  ~P x  -> 
y  e.  ~P U. A )
1211ssriv 3030 1  |-  U_ x  e.  A  ~P x  C_ 
~P U. A
Colors of variables: wff set class
Syntax hints:    e. wcel 1439   E.wrex 2361    C_ wss 3000   ~Pcpw 3433   U.cuni 3659   U_ciun 3736
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-in 3006  df-ss 3013  df-pw 3435  df-uni 3660  df-iun 3738
This theorem is referenced by: (None)
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