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Theorem iunn0 4375
Description: There is a nonempty class in an indexed collection  B ( x ) iff the indexed union of them is nonempty. (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 3126 . . 3  |-  ( E. x  e.  A  E. y  y  e.  B  <->  E. y E. x  e.  A  y  e.  B
)
2 eliun 4320 . . . 4  |-  ( y  e.  U_ x  e.  A  B  <->  E. x  e.  A  y  e.  B )
32exbii 1672 . . 3  |-  ( E. y  y  e.  U_ x  e.  A  B  <->  E. y E. x  e.  A  y  e.  B
)
41, 3bitr4i 252 . 2  |-  ( E. x  e.  A  E. y  y  e.  B  <->  E. y  y  e.  U_ x  e.  A  B
)
5 n0 3793 . . 3  |-  ( B  =/=  (/)  <->  E. y  y  e.  B )
65rexbii 2956 . 2  |-  ( E. x  e.  A  B  =/=  (/)  <->  E. x  e.  A  E. y  y  e.  B )
7 n0 3793 . 2  |-  ( U_ x  e.  A  B  =/=  (/)  <->  E. y  y  e. 
U_ x  e.  A  B )
84, 6, 73bitr4i 277 1  |-  ( E. x  e.  A  B  =/=  (/)  <->  U_ x  e.  A  B  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   E.wex 1617    e. wcel 1823    =/= wne 2649   E.wrex 2805   (/)c0 3783   U_ciun 4315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-v 3108  df-dif 3464  df-nul 3784  df-iun 4317
This theorem is referenced by:  fsuppmapnn0fiubex  12083  lbsextlem2  18003
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