Theorem intsng 3925

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 3652	. . 3 |- { A } = { A , A }
2	1	inteqi 3895	. 2 |- |^| { A } = |^| { A , A }
3		intprg 3924	. . . 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 3386	. . 3 |- ( A i^i A ) = A
6	4, 5	eqtrdi 2255	. 2 |- ( A e. V -> |^| { A , A } = A )
7	2, 6	eqtrid 2251	1 |- ( A e. V -> |^| { A } = A )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2177   i^i cin 3169   {csn 3638   {cpr 3639   |^|cint 3891

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-v 2775  df-un 3174  df-in 3176  df-sn 3644  df-pr 3645  df-int 3892

This theorem is referenced by:  intsn  3926  op1stbg  4534  riinint  4948