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Theorem intsng 4781
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 4449 . . 3 {𝐴} = {𝐴, 𝐴}
21inteqi 4750 . 2 {𝐴} = {𝐴, 𝐴}
3 intprg 4780 . . . 4 ((𝐴𝑉𝐴𝑉) → {𝐴, 𝐴} = (𝐴𝐴))
43anidms 559 . . 3 (𝐴𝑉 {𝐴, 𝐴} = (𝐴𝐴))
5 inidm 4077 . . 3 (𝐴𝐴) = 𝐴
64, 5syl6eq 2825 . 2 (𝐴𝑉 {𝐴, 𝐴} = 𝐴)
72, 6syl5eq 2821 1 (𝐴𝑉 {𝐴} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1508  wcel 2051  cin 3823  {csn 4436  {cpr 4438   cint 4746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2745
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ral 3088  df-rab 3092  df-v 3412  df-un 3829  df-in 3831  df-sn 4437  df-pr 4439  df-int 4747
This theorem is referenced by:  intsn  4782  riinint  5679  bj-snmoore  33949  elrfi  38720
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