Theorem intsng 3967

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 3687	. . 3 |- { A } = { A , A }
2	1	inteqi 3937	. 2 |- |^| { A } = |^| { A , A }
3		intprg 3966	. . . 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 3418	. . 3 |- ( A i^i A ) = A
6	4, 5	eqtrdi 2280	. 2 |- ( A e. V -> |^| { A , A } = A )
7	2, 6	eqtrid 2276	1 |- ( A e. V -> |^| { A } = A )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2202   i^i cin 3200   {csn 3673   {cpr 3674   |^|cint 3933

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-v 2805  df-un 3205  df-in 3207  df-sn 3679  df-pr 3680  df-int 3934

This theorem is referenced by:  intsn  3968  op1stbg  4582  riinint  4999