ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  intsng Unicode version

Theorem intsng 3717
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 3455 . . 3  |-  { A }  =  { A ,  A }
21inteqi 3687 . 2  |-  |^| { A }  =  |^| { A ,  A }
3 intprg 3716 . . . 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 3207 . . 3  |-  ( A  i^i  A )  =  A
64, 5syl6eq 2136 . 2  |-  ( A  e.  V  ->  |^| { A ,  A }  =  A )
72, 6syl5eq 2132 1  |-  ( A  e.  V  ->  |^| { A }  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    e. wcel 1438    i^i cin 2996   {csn 3441   {cpr 3442   |^|cint 3683
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-un 3001  df-in 3003  df-sn 3447  df-pr 3448  df-int 3684
This theorem is referenced by:  intsn  3718  op1stbg  4291  riinint  4682
  Copyright terms: Public domain W3C validator