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Theorem intsng 3934
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 3688 . . 3  |-  { A }  =  { A ,  A }
21inteqi 3903 . 2  |-  |^| { A }  =  |^| { A ,  A }
3 intprg 3933 . . . 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 3412 . . 3  |-  ( A  i^i  A )  =  A
64, 5syl6eq 2364 . 2  |-  ( A  e.  V  ->  |^| { A ,  A }  =  A )
72, 6syl5eq 2360 1  |-  ( A  e.  V  ->  |^| { A }  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1633    e. wcel 1701    i^i cin 3185   {csn 3674   {cpr 3675   |^|cint 3899
This theorem is referenced by:  intsn  3935  riinint  4972  elrfi  25917
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ral 2582  df-v 2824  df-un 3191  df-in 3193  df-sn 3680  df-pr 3681  df-int 3900
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