Theorem intunsn 3908

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 3901	. 2 |- |^| ( A u. { B } ) = ( |^| A i^i |^| { B } )
2		intunsn.1	. . . 4 |- B e. _V
3	2	intsn 3905	. . 3 |- |^| { B } = B
4	3	ineq2i 3357	. 2 |- ( |^| A i^i |^| { B } ) = ( |^| A i^i B )
5	1, 4	eqtri 2214	1 |- |^| ( A u. { B } ) = ( |^| A i^i B )

Syntax hints:   = wceq 1364   e. wcel 2164   _Vcvv 2760   u. cun 3151   i^i cin 3152   {csn 3618   |^|cint 3870

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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-v 2762  df-un 3157  df-in 3159  df-sn 3624  df-pr 3625  df-int 3871

This theorem is referenced by:  fiintim  6985