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Theorem intsng 3775
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 3511 . . 3 {𝐴} = {𝐴, 𝐴}
21inteqi 3745 . 2 {𝐴} = {𝐴, 𝐴}
3 intprg 3774 . . . 4 ((𝐴𝑉𝐴𝑉) → {𝐴, 𝐴} = (𝐴𝐴))
43anidms 394 . . 3 (𝐴𝑉 {𝐴, 𝐴} = (𝐴𝐴))
5 inidm 3255 . . 3 (𝐴𝐴) = 𝐴
64, 5syl6eq 2166 . 2 (𝐴𝑉 {𝐴, 𝐴} = 𝐴)
72, 6syl5eq 2162 1 (𝐴𝑉 {𝐴} = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1316  wcel 1465  cin 3040  {csn 3497  {cpr 3498   cint 3741
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-v 2662  df-un 3045  df-in 3047  df-sn 3503  df-pr 3504  df-int 3742
This theorem is referenced by:  intsn  3776  op1stbg  4370  riinint  4770
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