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Theorem intsng 3773
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 3509 . . 3 {𝐴} = {𝐴, 𝐴}
21inteqi 3743 . 2 {𝐴} = {𝐴, 𝐴}
3 intprg 3772 . . . 4 ((𝐴𝑉𝐴𝑉) → {𝐴, 𝐴} = (𝐴𝐴))
43anidms 392 . . 3 (𝐴𝑉 {𝐴, 𝐴} = (𝐴𝐴))
5 inidm 3253 . . 3 (𝐴𝐴) = 𝐴
64, 5syl6eq 2164 . 2 (𝐴𝑉 {𝐴, 𝐴} = 𝐴)
72, 6syl5eq 2160 1 (𝐴𝑉 {𝐴} = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1314  wcel 1463  cin 3038  {csn 3495  {cpr 3496   cint 3739
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-v 2660  df-un 3043  df-in 3045  df-sn 3501  df-pr 3502  df-int 3740
This theorem is referenced by:  intsn  3774  op1stbg  4368  riinint  4768
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