Theorem intsng 3880

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 3608	. . 3 |- { A } = { A , A }
2	1	inteqi 3850	. 2 |- |^| { A } = |^| { A , A }
3		intprg 3879	. . . 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 3346	. . 3 |- ( A i^i A ) = A
6	4, 5	eqtrdi 2226	. 2 |- ( A e. V -> |^| { A , A } = A )
7	2, 6	eqtrid 2222	1 |- ( A e. V -> |^| { A } = A )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   i^i cin 3130   {csn 3594   {cpr 3595   |^|cint 3846

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2741  df-un 3135  df-in 3137  df-sn 3600  df-pr 3601  df-int 3847

This theorem is referenced by:  intsn  3881  op1stbg  4481  riinint  4890