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Theorem intsng 3678
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 3420 . . 3 {𝐴} = {𝐴, 𝐴}
21inteqi 3648 . 2 {𝐴} = {𝐴, 𝐴}
3 intprg 3677 . . . 4 ((𝐴𝑉𝐴𝑉) → {𝐴, 𝐴} = (𝐴𝐴))
43anidms 389 . . 3 (𝐴𝑉 {𝐴, 𝐴} = (𝐴𝐴))
5 inidm 3182 . . 3 (𝐴𝐴) = 𝐴
64, 5syl6eq 2130 . 2 (𝐴𝑉 {𝐴, 𝐴} = 𝐴)
72, 6syl5eq 2126 1 (𝐴𝑉 {𝐴} = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  wcel 1434  cin 2973  {csn 3406  {cpr 3407   cint 3644
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-v 2604  df-un 2978  df-in 2980  df-sn 3412  df-pr 3413  df-int 3645
This theorem is referenced by:  intsn  3679  op1stbg  4236  riinint  4621
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