Theorem dfin5 3137

Description: Alternate definition for the intersection of two classes. (Contributed by NM, 6-Jul-2005.)

Assertion

Ref	Expression
dfin5	|- ( A i^i B ) = { x e. A | x e. B }

Distinct variable groups:   x, A   x, B

Proof of Theorem dfin5

Step	Hyp	Ref	Expression
1		df-in 3136	. 2 |- ( A i^i B ) = { x | ( x e. A /\ x e. B ) }
2		df-rab 2464	. 2 |- { x e. A | x e. B } = { x | ( x e. A /\ x e. B ) }
3	1, 2	eqtr4i 2201	1 |- ( A i^i B ) = { x e. A | x e. B }

Syntax hints:   /\ wa 104   = wceq 1353   e. wcel 2148   {cab 2163   {crab 2459   i^i cin 3129

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-5 1447  ax-gen 1449  ax-4 1510  ax-17 1526  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-rab 2464  df-in 3136

This theorem is referenced by:  nfin  3342  rabbi2dva  3344  ssfidc  6934  suprzubdc  11953  nninfdcex  11954  nnmindc  12035  nnminle  12036  znnen  12399  bj-inex  14662