Theorem intsng 3904

Description: Intersection of a singleton. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Assertion

Ref	Expression
intsng	|- ( A e. V -> |^| { A } = A )

Proof of Theorem intsng

Step	Hyp	Ref	Expression
1		dfsn2 3632	. . 3 |- { A } = { A , A }
2	1	inteqi 3874	. 2 |- |^| { A } = |^| { A , A }
3		intprg 3903	. . . 4 |- ( ( A e. V /\ A e. V ) -> |^| { A , A } = ( A i^i A ) )
4	3	anidms 397	. . 3 |- ( A e. V -> |^| { A , A } = ( A i^i A ) )
5		inidm 3368	. . 3 |- ( A i^i A ) = A
6	4, 5	eqtrdi 2242	. 2 |- ( A e. V -> |^| { A , A } = A )
7	2, 6	eqtrid 2238	1 |- ( A e. V -> |^| { A } = A )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164   i^i cin 3152   {csn 3618   {cpr 3619   |^|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:  intsn  3905  op1stbg  4510  riinint  4923