MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iunpwss Unicode version

Theorem iunpwss 3991
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 3944 . . 3  |-  ( E. x  e.  A  y 
C_  x  ->  y  C_ 
U_ x  e.  A  x )
2 eliun 3909 . . . 4  |-  ( y  e.  U_ x  e.  A  ~P x  <->  E. x  e.  A  y  e.  ~P x )
3 vex 2791 . . . . . 6  |-  y  e. 
_V
43elpw 3631 . . . . 5  |-  ( y  e.  ~P x  <->  y  C_  x )
54rexbii 2568 . . . 4  |-  ( E. x  e.  A  y  e.  ~P x  <->  E. x  e.  A  y  C_  x )
62, 5bitri 240 . . 3  |-  ( y  e.  U_ x  e.  A  ~P x  <->  E. x  e.  A  y  C_  x )
73elpw 3631 . . . 4  |-  ( y  e.  ~P U. A  <->  y 
C_  U. A )
8 uniiun 3955 . . . . 5  |-  U. A  =  U_ x  e.  A  x
98sseq2i 3203 . . . 4  |-  ( y 
C_  U. A  <->  y  C_  U_ x  e.  A  x )
107, 9bitri 240 . . 3  |-  ( y  e.  ~P U. A  <->  y 
C_  U_ x  e.  A  x )
111, 6, 103imtr4i 257 . 2  |-  ( y  e.  U_ x  e.  A  ~P x  -> 
y  e.  ~P U. A )
1211ssriv 3184 1  |-  U_ x  e.  A  ~P x  C_ 
~P U. A
Colors of variables: wff set class
Syntax hints:    e. wcel 1684   E.wrex 2544    C_ wss 3152   ~Pcpw 3625   U.cuni 3827   U_ciun 3905
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-in 3159  df-ss 3166  df-pw 3627  df-uni 3828  df-iun 3907
  Copyright terms: Public domain W3C validator