Theorem intsng 3874

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 3603	. . 3 |- { A } = { A , A }
2	1	inteqi 3844	. 2 |- |^| { A } = |^| { A , A }
3		intprg 3873	. . . 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 3342	. . 3 |- ( A i^i A ) = A
6	4, 5	eqtrdi 2224	. 2 |- ( A e. V -> |^| { A , A } = A )
7	2, 6	eqtrid 2220	1 |- ( A e. V -> |^| { A } = A )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2146   i^i cin 3126   {csn 3589   {cpr 3590   |^|cint 3840

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-v 2737  df-un 3131  df-in 3133  df-sn 3595  df-pr 3596  df-int 3841

This theorem is referenced by:  intsn  3875  op1stbg  4473  riinint  4881