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Theorem intsng 4644
 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 4327 . . 3 {𝐴} = {𝐴, 𝐴}
21inteqi 4613 . 2 {𝐴} = {𝐴, 𝐴}
3 intprg 4643 . . . 4 ((𝐴𝑉𝐴𝑉) → {𝐴, 𝐴} = (𝐴𝐴))
43anidms 548 . . 3 (𝐴𝑉 {𝐴, 𝐴} = (𝐴𝐴))
5 inidm 3969 . . 3 (𝐴𝐴) = 𝐴
64, 5syl6eq 2820 . 2 (𝐴𝑉 {𝐴, 𝐴} = 𝐴)
72, 6syl5eq 2816 1 (𝐴𝑉 {𝐴} = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1630   ∈ wcel 2144   ∩ cin 3720  {csn 4314  {cpr 4316  ∩ cint 4609 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-v 3351  df-un 3726  df-in 3728  df-sn 4315  df-pr 4317  df-int 4610 This theorem is referenced by:  intsn  4645  riinint  5520  bj-snmoore  33393  elrfi  37776
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