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Theorem intsng 3913
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
StepHypRef Expression
1 dfsn2 3667 . . 3  |-  { A }  =  { A ,  A }
21inteqi 3882 . 2  |-  |^| { A }  =  |^| { A ,  A }
3 intprg 3912 . . . 4  |-  ( ( A  e.  V  /\  A  e.  V )  ->  |^| { A ,  A }  =  ( A  i^i  A ) )
43anidms 626 . . 3  |-  ( A  e.  V  ->  |^| { A ,  A }  =  ( A  i^i  A ) )
5 inidm 3391 . . 3  |-  ( A  i^i  A )  =  A
64, 5syl6eq 2344 . 2  |-  ( A  e.  V  ->  |^| { A ,  A }  =  A )
72, 6syl5eq 2340 1  |-  ( A  e.  V  ->  |^| { A }  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696    i^i cin 3164   {csn 3653   {cpr 3654   |^|cint 3878
This theorem is referenced by:  intsn  3914  riinint  4951  moec  25150  elrfi  26872
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-v 2803  df-un 3170  df-in 3172  df-sn 3659  df-pr 3660  df-int 3879
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