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Theorem intsng 4873
Description: Intersection of a singleton. (Contributed by Stefan O'Rear, 22-Feb-2015.)
Assertion
Ref Expression
intsng (𝐴𝑉 {𝐴} = 𝐴)

Proof of Theorem intsng
StepHypRef Expression
1 dfsn2 4538 . . 3 {𝐴} = {𝐴, 𝐴}
21inteqi 4842 . 2 {𝐴} = {𝐴, 𝐴}
3 intprg 4872 . . . 4 ((𝐴𝑉𝐴𝑉) → {𝐴, 𝐴} = (𝐴𝐴))
43anidms 570 . . 3 (𝐴𝑉 {𝐴, 𝐴} = (𝐴𝐴))
5 inidm 4145 . . 3 (𝐴𝐴) = 𝐴
64, 5eqtrdi 2849 . 2 (𝐴𝑉 {𝐴, 𝐴} = 𝐴)
72, 6syl5eq 2845 1 (𝐴𝑉 {𝐴} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2111  cin 3880  {csn 4525  {cpr 4527   cint 4838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-12 2175  ax-ext 2770
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-rab 3115  df-v 3443  df-un 3886  df-in 3888  df-sn 4526  df-pr 4528  df-int 4839
This theorem is referenced by:  intsn  4874  riinint  5804  bj-snmoore  34528  bj-prmoore  34530  elrfi  39633
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