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Theorem intsng 4983
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 4639 . . 3 {𝐴} = {𝐴, 𝐴}
21inteqi 4950 . 2 {𝐴} = {𝐴, 𝐴}
3 intprg 4981 . . . 4 ((𝐴𝑉𝐴𝑉) → {𝐴, 𝐴} = (𝐴𝐴))
43anidms 566 . . 3 (𝐴𝑉 {𝐴, 𝐴} = (𝐴𝐴))
5 inidm 4227 . . 3 (𝐴𝐴) = 𝐴
64, 5eqtrdi 2793 . 2 (𝐴𝑉 {𝐴, 𝐴} = 𝐴)
72, 6eqtrid 2789 1 (𝐴𝑉 {𝐴} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  cin 3950  {csn 4626  {cpr 4628   cint 4946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-v 3482  df-un 3956  df-in 3958  df-sn 4627  df-pr 4629  df-int 4947
This theorem is referenced by:  intsn  4984  riinint  5982  bj-snmoore  37114  bj-prmoore  37116  elrfi  42705
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