Theorem iin0imm 4212

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

Step	Hyp	Ref	Expression
1		iinconstm 3936	1 |- ( E. y y e. A -> |^|_ x e. A (/) = (/) )

Syntax hints:   -> wi 4   = wceq 1373   E.wex 1515   e. wcel 2176   (/)c0 3460   |^|_ciin 3928

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-v 2774  df-iin 3930

This theorem is referenced by: (None)