Theorem iin0imm 4258

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 3979	1 |- ( E. y y e. A -> |^|_ x e. A (/) = (/) )

Syntax hints:   -> wi 4   = wceq 1397   E.wex 1540   e. wcel 2202   (/)c0 3494   |^|_ciin 3971

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-v 2804  df-iin 3973

This theorem is referenced by: (None)