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Theorem intsng 4109
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 3852 . . 3  |-  { A }  =  { A ,  A }
21inteqi 4078 . 2  |-  |^| { A }  =  |^| { A ,  A }
3 intprg 4108 . . . 4  |-  ( ( A  e.  V  /\  A  e.  V )  ->  |^| { A ,  A }  =  ( A  i^i  A ) )
43anidms 628 . . 3  |-  ( A  e.  V  ->  |^| { A ,  A }  =  ( A  i^i  A ) )
5 inidm 3535 . . 3  |-  ( A  i^i  A )  =  A
64, 5syl6eq 2490 . 2  |-  ( A  e.  V  ->  |^| { A ,  A }  =  A )
72, 6syl5eq 2486 1  |-  ( A  e.  V  ->  |^| { A }  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1727    i^i cin 3305   {csn 3838   {cpr 3839   |^|cint 4074
This theorem is referenced by:  intsn  4110  riinint  5155  elrfi  26786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ral 2716  df-v 2964  df-un 3311  df-in 3313  df-sn 3844  df-pr 3845  df-int 4075
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