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Theorem intsng 3705
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 3445 . . 3 {𝐴} = {𝐴, 𝐴}
21inteqi 3675 . 2 {𝐴} = {𝐴, 𝐴}
3 intprg 3704 . . . 4 ((𝐴𝑉𝐴𝑉) → {𝐴, 𝐴} = (𝐴𝐴))
43anidms 389 . . 3 (𝐴𝑉 {𝐴, 𝐴} = (𝐴𝐴))
5 inidm 3198 . . 3 (𝐴𝐴) = 𝐴
64, 5syl6eq 2133 . 2 (𝐴𝑉 {𝐴, 𝐴} = 𝐴)
72, 6syl5eq 2129 1 (𝐴𝑉 {𝐴} = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1287  wcel 1436  cin 2987  {csn 3431  {cpr 3432   cint 3671
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-v 2617  df-un 2992  df-in 2994  df-sn 3437  df-pr 3438  df-int 3672
This theorem is referenced by:  intsn  3706  op1stbg  4274  riinint  4662
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