Theorem intsng 3865

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 3597	. . 3 |- { A } = { A , A }
2	1	inteqi 3835	. 2 |- |^| { A } = |^| { A , A }
3		intprg 3864	. . . 4 |- ( ( A e. V /\ A e. V ) -> |^| { A , A } = ( A i^i A ) )
4	3	anidms 395	. . 3 |- ( A e. V -> |^| { A , A } = ( A i^i A ) )
5		inidm 3336	. . 3 |- ( A i^i A ) = A
6	4, 5	eqtrdi 2219	. 2 |- ( A e. V -> |^| { A , A } = A )
7	2, 6	eqtrid 2215	1 |- ( A e. V -> |^| { A } = A )

Syntax hints:   -> wi 4   = wceq 1348   e. wcel 2141   i^i cin 3120   {csn 3583   {cpr 3584   |^|cint 3831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-un 3125  df-in 3127  df-sn 3589  df-pr 3590  df-int 3832

This theorem is referenced by:  intsn  3866  op1stbg  4464  riinint  4872