Theorem in0 3279

Description: The intersection of a class with the empty set is the empty set. Theorem 16 of [Suppes] p. 26. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
in0	|- ( A i^i (/) ) = (/)

Proof of Theorem in0

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		noel 3255	. . . 4 |- -. x e. (/)
2	1	bianfi 888	. . 3 |- ( x e. (/) <-> ( x e. A /\ x e. (/) ) )
3	2	bicomi 130	. 2 |- ( ( x e. A /\ x e. (/) ) <-> x e. (/) )
4	3	ineqri 3159	1 |- ( A i^i (/) ) = (/)

Syntax hints:   /\ wa 102   = wceq 1284   e. wcel 1433   i^i cin 2972   (/)c0 3251

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-dif 2975  df-in 2979  df-nul 3252

This theorem is referenced by:  res0  4634