Theorem intsng 3805

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 3541	. . 3 |- { A } = { A , A }
2	1	inteqi 3775	. 2 |- |^| { A } = |^| { A , A }
3		intprg 3804	. . . 4 |- ( ( A e. V /\ A e. V ) -> |^| { A , A } = ( A i^i A ) )
4	3	anidms 394	. . 3 |- ( A e. V -> |^| { A , A } = ( A i^i A ) )
5		inidm 3285	. . 3 |- ( A i^i A ) = A
6	4, 5	syl6eq 2188	. 2 |- ( A e. V -> |^| { A , A } = A )
7	2, 6	syl5eq 2184	1 |- ( A e. V -> |^| { A } = A )

Syntax hints:   -> wi 4   = wceq 1331   e. wcel 1480   i^i cin 3070   {csn 3527   {cpr 3528   |^|cint 3771

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-un 3075  df-in 3077  df-sn 3533  df-pr 3534  df-int 3772

This theorem is referenced by:  intsn  3806  op1stbg  4400  riinint  4800