Theorem in0 3321

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 3291	. . . 4 |- -. x e. (/)
2	1	bianfi 894	. . 3 |- ( x e. (/) <-> ( x e. A /\ x e. (/) ) )
3	2	bicomi 131	. 2 |- ( ( x e. A /\ x e. (/) ) <-> x e. (/) )
4	3	ineqri 3194	1 |- ( A i^i (/) ) = (/)

Syntax hints:   /\ wa 103   = wceq 1290   e. wcel 1439   i^i cin 2999   (/)c0 3287

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-dif 3002  df-in 3006  df-nul 3288

This theorem is referenced by:  0in  3322  res0  4730