Theorem intsng 3909

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 3637	. . 3 |- { A } = { A , A }
2	1	inteqi 3879	. 2 |- |^| { A } = |^| { A , A }
3		intprg 3908	. . . 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 3373	. . 3 |- ( A i^i A ) = A
6	4, 5	eqtrdi 2245	. 2 |- ( A e. V -> |^| { A , A } = A )
7	2, 6	eqtrid 2241	1 |- ( A e. V -> |^| { A } = A )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167   i^i cin 3156   {csn 3623   {cpr 3624   |^|cint 3875

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-un 3161  df-in 3163  df-sn 3629  df-pr 3630  df-int 3876

This theorem is referenced by:  intsn  3910  op1stbg  4515  riinint  4928