Theorem intsn 4991

Description: The intersection of a singleton is its member. Theorem 70 of [Suppes] p. 41. (Contributed by NM, 29-Sep-2002.)

Hypothesis

Ref	Expression
intsn.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
intsn	⊢ ∩ {𝐴} = 𝐴

Proof of Theorem intsn

Step	Hyp	Ref	Expression
1		intsn.1	. 2 ⊢ 𝐴 ∈ V
2		intsng 4990	. 2 ⊢ (𝐴 ∈ V → ∩ {𝐴} = 𝐴)
3	1, 2	ax-mp 5	1 ⊢ ∩ {𝐴} = 𝐴

Syntax hints:   = wceq 1542   ∈ wcel 2107  Vcvv 3475  {csn 4629  ∩ cint 4951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-v 3477  df-un 3954  df-in 3956  df-sn 4630  df-pr 4632  df-int 4952

This theorem is referenced by:  uniintsn  4992  intunsn  4994  op1stb  5472  op2ndb  6227  ssfii  9414  cf0  10246  cflecard  10248  uffix  23425  iotain  43176