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Theorem iunpwss 4408
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 4357 . . 3  |-  ( E. x  e.  A  y 
C_  x  ->  y  C_ 
U_ x  e.  A  x )
2 eliun 4320 . . . 4  |-  ( y  e.  U_ x  e.  A  ~P x  <->  E. x  e.  A  y  e.  ~P x )
3 selpw 4006 . . . . 5  |-  ( y  e.  ~P x  <->  y  C_  x )
43rexbii 2956 . . . 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 4006 . . . 4  |-  ( y  e.  ~P U. A  <->  y 
C_  U. A )
7 uniiun 4368 . . . . 5  |-  U. A  =  U_ x  e.  A  x
87sseq2i 3514 . . . 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 3493 1  |-  U_ x  e.  A  ~P x  C_ 
~P U. A
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1823   E.wrex 2805    C_ wss 3461   ~Pcpw 3999   U.cuni 4235   U_ciun 4315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-v 3108  df-in 3468  df-ss 3475  df-pw 4001  df-uni 4236  df-iun 4317
This theorem is referenced by: (None)
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