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Theorem ssiun3 23962
Description: Subset equivalence for an indexed union. (Contributed by Thierry Arnoux, 17-Oct-2016.)
Assertion
Ref Expression
ssiun3  |-  ( A. y  e.  C  E. x  e.  A  y  e.  B  <->  C  C_  U_ x  e.  A  B )
Distinct variable groups:    x, y    y, A    y, B    y, C
Allowed substitution hints:    A( x)    B( x)    C( x)

Proof of Theorem ssiun3
StepHypRef Expression
1 dfss2 3297 . 2  |-  ( C 
C_  U_ x  e.  A  B 
<-> 
A. y ( y  e.  C  ->  y  e.  U_ x  e.  A  B ) )
2 df-ral 2671 . 2  |-  ( A. y  e.  C  y  e.  U_ x  e.  A  B 
<-> 
A. y ( y  e.  C  ->  y  e.  U_ x  e.  A  B ) )
3 eliun 4057 . . 3  |-  ( y  e.  U_ x  e.  A  B  <->  E. x  e.  A  y  e.  B )
43ralbii 2690 . 2  |-  ( A. y  e.  C  y  e.  U_ x  e.  A  B 
<-> 
A. y  e.  C  E. x  e.  A  y  e.  B )
51, 2, 43bitr2ri 266 1  |-  ( A. y  e.  C  E. x  e.  A  y  e.  B  <->  C  C_  U_ x  e.  A  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1546    e. wcel 1721   A.wral 2666   E.wrex 2667    C_ wss 3280   U_ciun 4053
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rex 2672  df-v 2918  df-in 3287  df-ss 3294  df-iun 4055
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