Theorem intsng 3771

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 3507	. . 3 |- { A } = { A , A }
2	1	inteqi 3741	. 2 |- |^| { A } = |^| { A , A }
3		intprg 3770	. . . 4 |- ( ( A e. V /\ A e. V ) -> |^| { A , A } = ( A i^i A ) )
4	3	anidms 392	. . 3 |- ( A e. V -> |^| { A , A } = ( A i^i A ) )
5		inidm 3251	. . 3 |- ( A i^i A ) = A
6	4, 5	syl6eq 2163	. 2 |- ( A e. V -> |^| { A , A } = A )
7	2, 6	syl5eq 2159	1 |- ( A e. V -> |^| { A } = A )

Syntax hints:   -> wi 4   = wceq 1314   e. wcel 1463   i^i cin 3036   {csn 3493   {cpr 3494   |^|cint 3737

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-v 2659  df-un 3041  df-in 3043  df-sn 3499  df-pr 3500  df-int 3738

This theorem is referenced by:  intsn  3772  op1stbg  4360  riinint  4758