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Theorem iunn0 4115
Description: There is a non-empty class in an indexed collection  B ( x ) iff the indexed union of them is non-empty. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
iunn0  |-  ( E. x  e.  A  B  =/=  (/)  <->  U_ x  e.  A  B  =/=  (/) )
Distinct variable group:    x, A
Allowed substitution hint:    B( x)

Proof of Theorem iunn0
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 rexcom4 2939 . . 3  |-  ( E. x  e.  A  E. y  y  e.  B  <->  E. y E. x  e.  A  y  e.  B
)
2 eliun 4061 . . . 4  |-  ( y  e.  U_ x  e.  A  B  <->  E. x  e.  A  y  e.  B )
32exbii 1589 . . 3  |-  ( E. y  y  e.  U_ x  e.  A  B  <->  E. y E. x  e.  A  y  e.  B
)
41, 3bitr4i 244 . 2  |-  ( E. x  e.  A  E. y  y  e.  B  <->  E. y  y  e.  U_ x  e.  A  B
)
5 n0 3601 . . 3  |-  ( B  =/=  (/)  <->  E. y  y  e.  B )
65rexbii 2695 . 2  |-  ( E. x  e.  A  B  =/=  (/)  <->  E. x  e.  A  E. y  y  e.  B )
7 n0 3601 . 2  |-  ( U_ x  e.  A  B  =/=  (/)  <->  E. y  y  e. 
U_ x  e.  A  B )
84, 6, 73bitr4i 269 1  |-  ( E. x  e.  A  B  =/=  (/)  <->  U_ x  e.  A  B  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   E.wex 1547    e. wcel 1721    =/= wne 2571   E.wrex 2671   (/)c0 3592   U_ciun 4057
This theorem is referenced by:  lbsextlem2  16190
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 2389
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 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-v 2922  df-dif 3287  df-nul 3593  df-iun 4059
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