Theorem in0 3443

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 3413	. . . 4 |- -. x e. (/)
2	1	bianfi 937	. . 3 |- ( x e. (/) <-> ( x e. A /\ x e. (/) ) )
3	2	bicomi 131	. 2 |- ( ( x e. A /\ x e. (/) ) <-> x e. (/) )
4	3	ineqri 3315	1 |- ( A i^i (/) ) = (/)

Syntax hints:   /\ wa 103   = wceq 1343   e. wcel 2136   i^i cin 3115   (/)c0 3409

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-dif 3118  df-in 3122  df-nul 3410

This theorem is referenced by:  0in  3444  res0  4888  dju0en  7170  rest0  12819