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Theorem iunn0 3962
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 2807 . . 3  |-  ( E. x  e.  A  E. y  y  e.  B  <->  E. y E. x  e.  A  y  e.  B
)
2 eliun 3909 . . . 4  |-  ( y  e.  U_ x  e.  A  B  <->  E. x  e.  A  y  e.  B )
32exbii 1569 . . 3  |-  ( E. y  y  e.  U_ x  e.  A  B  <->  E. y E. x  e.  A  y  e.  B
)
41, 3bitr4i 243 . 2  |-  ( E. x  e.  A  E. y  y  e.  B  <->  E. y  y  e.  U_ x  e.  A  B
)
5 n0 3464 . . 3  |-  ( B  =/=  (/)  <->  E. y  y  e.  B )
65rexbii 2568 . 2  |-  ( E. x  e.  A  B  =/=  (/)  <->  E. x  e.  A  E. y  y  e.  B )
7 n0 3464 . 2  |-  ( U_ x  e.  A  B  =/=  (/)  <->  E. y  y  e. 
U_ x  e.  A  B )
84, 6, 73bitr4i 268 1  |-  ( E. x  e.  A  B  =/=  (/)  <->  U_ x  e.  A  B  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   E.wex 1528    e. wcel 1684    =/= wne 2446   E.wrex 2544   (/)c0 3455   U_ciun 3905
This theorem is referenced by:  lbsextlem2  15912
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-ne 2448  df-ral 2548  df-rex 2549  df-v 2790  df-dif 3155  df-nul 3456  df-iun 3907
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