Theorem intunsn 3817

Description: Theorem joining a singleton to an intersection. (Contributed by NM, 29-Sep-2002.)

Hypothesis

Ref	Expression
intunsn.1	|- B e. _V

Assertion

Ref	Expression
intunsn	|- |^| ( A u. { B } ) = ( |^| A i^i B )

Proof of Theorem intunsn

Step	Hyp	Ref	Expression
1		intun 3810	. 2 |- |^| ( A u. { B } ) = ( |^| A i^i |^| { B } )
2		intunsn.1	. . . 4 |- B e. _V
3	2	intsn 3814	. . 3 |- |^| { B } = B
4	3	ineq2i 3279	. 2 |- ( |^| A i^i |^| { B } ) = ( |^| A i^i B )
5	1, 4	eqtri 2161	1 |- |^| ( A u. { B } ) = ( |^| A i^i B )

Syntax hints:   = wceq 1332   e. wcel 1481   _Vcvv 2689   u. cun 3074   i^i cin 3075   {csn 3532   |^|cint 3779

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-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2691  df-un 3080  df-in 3082  df-sn 3538  df-pr 3539  df-int 3780

This theorem is referenced by:  fiintim  6825