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Theorem iin0imm 4062
Description: An indexed intersection of the empty set, with an inhabited index set, is empty. (Contributed by Jim Kingdon, 29-Aug-2018.)
Assertion
Ref Expression
iin0imm  |-  ( E. y  y  e.  A  -> 
|^|_ x  e.  A  (/)  =  (/) )
Distinct variable groups:    y, A    x, A

Proof of Theorem iin0imm
StepHypRef Expression
1 iinconstm 3792 1  |-  ( E. y  y  e.  A  -> 
|^|_ x  e.  A  (/)  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316   E.wex 1453    e. wcel 1465   (/)c0 3333   |^|_ciin 3784
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-v 2662  df-iin 3786
This theorem is referenced by: (None)
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