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Theorem intsng 4902
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 4570 . . 3 {𝐴} = {𝐴, 𝐴}
21inteqi 4871 . 2 {𝐴} = {𝐴, 𝐴}
3 intprg 4901 . . . 4 ((𝐴𝑉𝐴𝑉) → {𝐴, 𝐴} = (𝐴𝐴))
43anidms 567 . . 3 (𝐴𝑉 {𝐴, 𝐴} = (𝐴𝐴))
5 inidm 4192 . . 3 (𝐴𝐴) = 𝐴
64, 5syl6eq 2869 . 2 (𝐴𝑉 {𝐴, 𝐴} = 𝐴)
72, 6syl5eq 2865 1 (𝐴𝑉 {𝐴} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  cin 3932  {csn 4557  {cpr 4559   cint 4867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rab 3144  df-v 3494  df-un 3938  df-in 3940  df-sn 4558  df-pr 4560  df-int 4868
This theorem is referenced by:  intsn  4903  riinint  5832  bj-snmoore  34299  elrfi  39169
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