Theorem intunsn 4155

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 4148	. 2 |- |^| ( A u. { B } ) = ( |^| A i^i |^| { B } )
2		intunsn.1	. . . 4 |- B e. _V
3	2	intsn 4152	. . 3 |- |^| { B } = B
4	3	ineq2i 3537	. 2 |- ( |^| A i^i |^| { B } ) = ( |^| A i^i B )
5	1, 4	eqtri 2453	1 |- |^| ( A u. { B } ) = ( |^| A i^i B )

Syntax hints:   = wceq 1362   e. wcel 1755   _Vcvv 2962   u. cun 3314   i^i cin 3315   {csn 3865   |^|cint 4116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ral 2710  df-v 2964  df-un 3321  df-in 3323  df-sn 3866  df-pr 3868  df-int 4117

This theorem is referenced by:  fiint  7576  incexclem  13282  heibor1lem  28552