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Theorem intsng 3841
 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 3574 . . 3 {𝐴} = {𝐴, 𝐴}
21inteqi 3811 . 2 {𝐴} = {𝐴, 𝐴}
3 intprg 3840 . . . 4 ((𝐴𝑉𝐴𝑉) → {𝐴, 𝐴} = (𝐴𝐴))
43anidms 395 . . 3 (𝐴𝑉 {𝐴, 𝐴} = (𝐴𝐴))
5 inidm 3316 . . 3 (𝐴𝐴) = 𝐴
64, 5eqtrdi 2206 . 2 (𝐴𝑉 {𝐴, 𝐴} = 𝐴)
72, 6syl5eq 2202 1 (𝐴𝑉 {𝐴} = 𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1335   ∈ wcel 2128   ∩ cin 3101  {csn 3560  {cpr 3561  ∩ cint 3807 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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714  df-un 3106  df-in 3108  df-sn 3566  df-pr 3567  df-int 3808 This theorem is referenced by:  intsn  3842  op1stbg  4437  riinint  4844
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