Theorem iin0imm 4228

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 3950	1 |- ( E. y y e. A -> |^|_ x e. A (/) = (/) )

Syntax hints:   -> wi 4   = wceq 1373   E.wex 1516   e. wcel 2178   (/)c0 3468   |^|_ciin 3942

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-v 2778  df-iin 3944

This theorem is referenced by: (None)