Theorem intunsn 4267

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 4260	. 2 |- |^| ( A u. { B } ) = ( |^| A i^i |^| { B } )
2		intunsn.1	. . . 4 |- B e. _V
3	2	intsn 4264	. . 3 |- |^| { B } = B
4	3	ineq2i 3638	. 2 |- ( |^| A i^i |^| { B } ) = ( |^| A i^i B )
5	1, 4	eqtri 2431	1 |- |^| ( A u. { B } ) = ( |^| A i^i B )

Syntax hints:   = wceq 1405   e. wcel 1842   _Vcvv 3059   u. cun 3412   i^i cin 3413   {csn 3972   |^|cint 4227

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2759  df-v 3061  df-un 3419  df-in 3421  df-sn 3973  df-pr 3975  df-int 4228

This theorem is referenced by:  fiint  7831  incexclem  13799  heibor1lem  31587