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Theorem iundif2ss 3848
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 3050 . . . . . 6  |-  ( y  e.  ( B  \  C )  <->  ( y  e.  B  /\  -.  y  e.  C ) )
21rexbii 2419 . . . . 5  |-  ( E. x  e.  A  y  e.  ( B  \  C )  <->  E. x  e.  A  ( y  e.  B  /\  -.  y  e.  C ) )
3 r19.42v 2565 . . . . 5  |-  ( E. x  e.  A  ( y  e.  B  /\  -.  y  e.  C
)  <->  ( y  e.  B  /\  E. x  e.  A  -.  y  e.  C ) )
42, 3bitri 183 . . . 4  |-  ( E. x  e.  A  y  e.  ( B  \  C )  <->  ( y  e.  B  /\  E. x  e.  A  -.  y  e.  C ) )
5 rexnalim 2404 . . . . . 6  |-  ( E. x  e.  A  -.  y  e.  C  ->  -. 
A. x  e.  A  y  e.  C )
6 vex 2663 . . . . . . 7  |-  y  e. 
_V
7 eliin 3788 . . . . . . 7  |-  ( y  e.  _V  ->  (
y  e.  |^|_ x  e.  A  C  <->  A. x  e.  A  y  e.  C ) )
86, 7ax-mp 5 . . . . . 6  |-  ( y  e.  |^|_ x  e.  A  C 
<-> 
A. x  e.  A  y  e.  C )
95, 8sylnibr 651 . . . . 5  |-  ( E. x  e.  A  -.  y  e.  C  ->  -.  y  e.  |^|_ x  e.  A  C )
109anim2i 339 . . . 4  |-  ( ( y  e.  B  /\  E. x  e.  A  -.  y  e.  C )  ->  ( y  e.  B  /\  -.  y  e.  |^|_ x  e.  A  C ) )
114, 10sylbi 120 . . 3  |-  ( E. x  e.  A  y  e.  ( B  \  C )  ->  (
y  e.  B  /\  -.  y  e.  |^|_ x  e.  A  C )
)
12 eliun 3787 . . 3  |-  ( y  e.  U_ x  e.  A  ( B  \  C )  <->  E. x  e.  A  y  e.  ( B  \  C ) )
13 eldif 3050 . . 3  |-  ( y  e.  ( B  \  |^|_ x  e.  A  C
)  <->  ( y  e.  B  /\  -.  y  e.  |^|_ x  e.  A  C ) )
1411, 12, 133imtr4i 200 . 2  |-  ( y  e.  U_ x  e.  A  ( B  \  C )  ->  y  e.  ( B  \  |^|_ x  e.  A  C ) )
1514ssriv 3071 1  |-  U_ x  e.  A  ( B  \  C )  C_  ( B  \  |^|_ x  e.  A  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    <-> wb 104    e. wcel 1465   A.wral 2393   E.wrex 2394   _Vcvv 2660    \ cdif 3038    C_ wss 3041   U_ciun 3783   |^|_ciin 3784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-dif 3043  df-in 3047  df-ss 3054  df-iun 3785  df-iin 3786
This theorem is referenced by: (None)
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