Theorem intsng 3956

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 3680	. . 3 |- { A } = { A , A }
2	1	inteqi 3926	. 2 |- |^| { A } = |^| { A , A }
3		intprg 3955	. . . 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 3413	. . 3 |- ( A i^i A ) = A
6	4, 5	eqtrdi 2278	. 2 |- ( A e. V -> |^| { A , A } = A )
7	2, 6	eqtrid 2274	1 |- ( A e. V -> |^| { A } = A )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   i^i cin 3196   {csn 3666   {cpr 3667   |^|cint 3922

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2801  df-un 3201  df-in 3203  df-sn 3672  df-pr 3673  df-int 3923

This theorem is referenced by:  intsn  3957  op1stbg  4569  riinint  4984