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Theorem iunn0m 3843
Description: There is an inhabited class in an indexed collection  B ( x ) iff the indexed union of them is inhabited. (Contributed by Jim Kingdon, 16-Aug-2018.)
Assertion
Ref Expression
iunn0m  |-  ( E. x  e.  A  E. y  y  e.  B  <->  E. y  y  e.  U_ x  e.  A  B
)
Distinct variable groups:    x, y, A   
y, B
Allowed substitution hint:    B( x)

Proof of Theorem iunn0m
StepHypRef Expression
1 rexcom4 2683 . 2  |-  ( E. x  e.  A  E. y  y  e.  B  <->  E. y E. x  e.  A  y  e.  B
)
2 eliun 3787 . . 3  |-  ( y  e.  U_ x  e.  A  B  <->  E. x  e.  A  y  e.  B )
32exbii 1569 . 2  |-  ( E. y  y  e.  U_ x  e.  A  B  <->  E. y E. x  e.  A  y  e.  B
)
41, 3bitr4i 186 1  |-  ( E. x  e.  A  E. y  y  e.  B  <->  E. y  y  e.  U_ x  e.  A  B
)
Colors of variables: wff set class
Syntax hints:    <-> wb 104   E.wex 1453    e. wcel 1465   E.wrex 2394   U_ciun 3783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-iun 3785
This theorem is referenced by: (None)
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