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Theorem iundif2ss 3763
Description: Indexed union of class difference. Compare to theorem "De Morgan's laws" in [Enderton] p. 31. (Contributed by Jim Kingdon, 17-Aug-2018.)
Assertion
Ref Expression
iundif2ss  |-  U_ x  e.  A  ( B  \  C )  C_  ( B  \  |^|_ x  e.  A  C )
Distinct variable group:    x, B
Allowed substitution hints:    A( x)    C( x)

Proof of Theorem iundif2ss
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eldif 2991 . . . . . 6  |-  ( y  e.  ( B  \  C )  <->  ( y  e.  B  /\  -.  y  e.  C ) )
21rexbii 2378 . . . . 5  |-  ( E. x  e.  A  y  e.  ( B  \  C )  <->  E. x  e.  A  ( y  e.  B  /\  -.  y  e.  C ) )
3 r19.42v 2516 . . . . 5  |-  ( E. x  e.  A  ( y  e.  B  /\  -.  y  e.  C
)  <->  ( y  e.  B  /\  E. x  e.  A  -.  y  e.  C ) )
42, 3bitri 182 . . . 4  |-  ( E. x  e.  A  y  e.  ( B  \  C )  <->  ( y  e.  B  /\  E. x  e.  A  -.  y  e.  C ) )
5 rexnalim 2364 . . . . . 6  |-  ( E. x  e.  A  -.  y  e.  C  ->  -. 
A. x  e.  A  y  e.  C )
6 vex 2613 . . . . . . 7  |-  y  e. 
_V
7 eliin 3703 . . . . . . 7  |-  ( y  e.  _V  ->  (
y  e.  |^|_ x  e.  A  C  <->  A. x  e.  A  y  e.  C ) )
86, 7ax-mp 7 . . . . . 6  |-  ( y  e.  |^|_ x  e.  A  C 
<-> 
A. x  e.  A  y  e.  C )
95, 8sylnibr 635 . . . . 5  |-  ( E. x  e.  A  -.  y  e.  C  ->  -.  y  e.  |^|_ x  e.  A  C )
109anim2i 334 . . . 4  |-  ( ( y  e.  B  /\  E. x  e.  A  -.  y  e.  C )  ->  ( y  e.  B  /\  -.  y  e.  |^|_ x  e.  A  C ) )
114, 10sylbi 119 . . 3  |-  ( E. x  e.  A  y  e.  ( B  \  C )  ->  (
y  e.  B  /\  -.  y  e.  |^|_ x  e.  A  C )
)
12 eliun 3702 . . 3  |-  ( y  e.  U_ x  e.  A  ( B  \  C )  <->  E. x  e.  A  y  e.  ( B  \  C ) )
13 eldif 2991 . . 3  |-  ( y  e.  ( B  \  |^|_ x  e.  A  C
)  <->  ( y  e.  B  /\  -.  y  e.  |^|_ x  e.  A  C ) )
1411, 12, 133imtr4i 199 . 2  |-  ( y  e.  U_ x  e.  A  ( B  \  C )  ->  y  e.  ( B  \  |^|_ x  e.  A  C ) )
1514ssriv 3012 1  |-  U_ x  e.  A  ( B  \  C )  C_  ( B  \  |^|_ x  e.  A  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 102    <-> wb 103    e. wcel 1434   A.wral 2353   E.wrex 2354   _Vcvv 2610    \ cdif 2979    C_ wss 2982   U_ciun 3698   |^|_ciin 3699
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-dif 2984  df-in 2988  df-ss 2995  df-iun 3700  df-iin 3701
This theorem is referenced by: (None)
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