Theorem intunsn 4516

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 4509	. 2 |- |^| ( A u. { B } ) = ( |^| A i^i |^| { B } )
2		intunsn.1	. . . 4 |- B e. _V
3	2	intsn 4513	. . 3 |- |^| { B } = B
4	3	ineq2i 3811	. 2 |- ( |^| A i^i |^| { B } ) = ( |^| A i^i B )
5	1, 4	eqtri 2644	1 |- |^| ( A u. { B } ) = ( |^| A i^i B )

Syntax hints:   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   i^i cin 3573   {csn 4177   |^|cint 4475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-un 3579  df-in 3581  df-sn 4178  df-pr 4180  df-int 4476

This theorem is referenced by:  fiint  8237  incexclem  14568  heibor1lem  33608