Theorem intsng 3649

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 3389	. . 3 |- { A } = { A , A }
2	1	inteqi 3619	. 2 |- |^| { A } = |^| { A , A }
3		intprg 3648	. . . 4 |- ( ( A e. V /\ A e. V ) -> |^| { A , A } = ( A i^i A ) )
4	3	anidms 377	. . 3 |- ( A e. V -> |^| { A , A } = ( A i^i A ) )
5		inidm 3146	. . 3 |- ( A i^i A ) = A
6	4, 5	syl6eq 2088	. 2 |- ( A e. V -> |^| { A , A } = A )
7	2, 6	syl5eq 2084	1 |- ( A e. V -> |^| { A } = A )

Syntax hints:   -> wi 4   = wceq 1243   e. wcel 1393   i^i cin 2916   {csn 3375   {cpr 3376   |^|cint 3615

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-un 2922  df-in 2924  df-sn 3381  df-pr 3382  df-int 3616

This theorem is referenced by:  intsn  3650  op1stbg  4210  riinint  4593