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Theorem iunn0 4153
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 2977 . . 3  |-  ( E. x  e.  A  E. y  y  e.  B  <->  E. y E. x  e.  A  y  e.  B
)
2 eliun 4099 . . . 4  |-  ( y  e.  U_ x  e.  A  B  <->  E. x  e.  A  y  e.  B )
32exbii 1593 . . 3  |-  ( E. y  y  e.  U_ x  e.  A  B  <->  E. y E. x  e.  A  y  e.  B
)
41, 3bitr4i 245 . 2  |-  ( E. x  e.  A  E. y  y  e.  B  <->  E. y  y  e.  U_ x  e.  A  B
)
5 n0 3639 . . 3  |-  ( B  =/=  (/)  <->  E. y  y  e.  B )
65rexbii 2732 . 2  |-  ( E. x  e.  A  B  =/=  (/)  <->  E. x  e.  A  E. y  y  e.  B )
7 n0 3639 . 2  |-  ( U_ x  e.  A  B  =/=  (/)  <->  E. y  y  e. 
U_ x  e.  A  B )
84, 6, 73bitr4i 270 1  |-  ( E. x  e.  A  B  =/=  (/)  <->  U_ x  e.  A  B  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178   E.wex 1551    e. wcel 1726    =/= wne 2601   E.wrex 2708   (/)c0 3630   U_ciun 4095
This theorem is referenced by:  lbsextlem2  16236
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-v 2960  df-dif 3325  df-nul 3631  df-iun 4097
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