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Theorem intsng 3691
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 3431 . . 3  |-  { A }  =  { A ,  A }
21inteqi 3661 . 2  |-  |^| { A }  =  |^| { A ,  A }
3 intprg 3690 . . . 4  |-  ( ( A  e.  V  /\  A  e.  V )  ->  |^| { A ,  A }  =  ( A  i^i  A ) )
43anidms 389 . . 3  |-  ( A  e.  V  ->  |^| { A ,  A }  =  ( A  i^i  A ) )
5 inidm 3192 . . 3  |-  ( A  i^i  A )  =  A
64, 5syl6eq 2131 . 2  |-  ( A  e.  V  ->  |^| { A ,  A }  =  A )
72, 6syl5eq 2127 1  |-  ( A  e.  V  ->  |^| { A }  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    e. wcel 1434    i^i cin 2982   {csn 3417   {cpr 3418   |^|cint 3657
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2613  df-un 2987  df-in 2989  df-sn 3423  df-pr 3424  df-int 3658
This theorem is referenced by:  intsn  3692  op1stbg  4257  riinint  4642
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