Theorem intsng 3999

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 3719	. . 3 |- { A } = { A , A }
2	1	inteqi 3969	. 2 |- |^| { A } = |^| { A , A }
3		intprg 3998	. . . 4 |- ( ( A e. V /\ A e. V ) -> |^| { A , A } = ( A i^i A ) )
4	3	anidms 401	. . 3 |- ( A e. V -> |^| { A , A } = ( A i^i A ) )
5		inidm 3440	. . 3 |- ( A i^i A ) = A
6	4, 5	eqtrdi 2287	. 2 |- ( A e. V -> |^| { A , A } = A )
7	2, 6	eqtrid 2283	1 |- ( A e. V -> |^| { A } = A )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   i^i cin 3219   {csn 3705   {cpr 3706   |^|cint 3965

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-un 3224  df-in 3226  df-sn 3711  df-pr 3712  df-int 3966

This theorem is referenced by:  intsn  4000  op1stbg  4620  riinint  5038