Theorem in0 3557

Description: The intersection of a class with the empty set is the empty set. Dual of unv 3560. Commuted form of 0in 3558. 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 3525	. . . 4 |- -. x e. (/)
2	1	bianfi 960	. . 3 |- ( x e. (/) <-> ( x e. A /\ x e. (/) ) )
3	2	bicomi 132	. 2 |- ( ( x e. A /\ x e. (/) ) <-> x e. (/) )
4	3	ineqri 3424	1 |- ( A i^i (/) ) = (/)

Syntax hints:   /\ wa 104   = wceq 1402   e. wcel 2209   i^i cin 3219   (/)c0 3520

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-dif 3222  df-in 3226  df-nul 3521

This theorem is referenced by:  0in  3558  res0  5062  dju0en  7560  bitsinv1  12707  ballotfilemfval0  13213  rest0  15203