Theorem intsng 3691

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 3431	. . 3 |- { A } = { A , A }
2	1	inteqi 3661	. 2 |- |^| { A } = |^| { A , A }
3		intprg 3690	. . . 4 |- ( ( A e. V /\ A e. V ) -> |^| { A , A } = ( A i^i A ) )
4	3	anidms 389	. . 3 |- ( A e. V -> |^| { A , A } = ( A i^i A ) )
5		inidm 3192	. . 3 |- ( A i^i A ) = A
6	4, 5	syl6eq 2131	. 2 |- ( A e. V -> |^| { A , A } = A )
7	2, 6	syl5eq 2127	1 |- ( A e. V -> |^| { A } = A )

Syntax hints:   -> wi 4   = wceq 1285   e. wcel 1434   i^i cin 2982   {csn 3417   {cpr 3418   |^|cint 3657

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2613  df-un 2987  df-in 2989  df-sn 3423  df-pr 3424  df-int 3658

This theorem is referenced by:  intsn  3692  op1stbg  4257  riinint  4642