Theorem iin0imm 4201

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 3925	1 |- ( E. y y e. A -> |^|_ x e. A (/) = (/) )

Syntax hints:   -> wi 4   = wceq 1364   E.wex 1506   e. wcel 2167   (/)c0 3450   |^|_ciin 3917

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-iin 3919

This theorem is referenced by: (None)