Theorem in0 3531

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 3500	. . . 4 |- -. x e. (/)
2	1	bianfi 956	. . 3 |- ( x e. (/) <-> ( x e. A /\ x e. (/) ) )
3	2	bicomi 132	. 2 |- ( ( x e. A /\ x e. (/) ) <-> x e. (/) )
4	3	ineqri 3402	1 |- ( A i^i (/) ) = (/)

Syntax hints:   /\ wa 104   = wceq 1398   e. wcel 2202   i^i cin 3200   (/)c0 3496

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-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-dif 3203  df-in 3207  df-nul 3497

This theorem is referenced by:  0in  3532  res0  5023  dju0en  7489  bitsinv1  12603  rest0  14990