Theorem intsn 5008

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 5007	. 2 ⊢ (𝐴 ∈ V → ∩ {𝐴} = 𝐴)
3	1, 2	ax-mp 5	1 ⊢ ∩ {𝐴} = 𝐴

Syntax hints:   = wceq 1537   ∈ wcel 2108  Vcvv 3488  {csn 4648  ∩ cint 4970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-v 3490  df-un 3981  df-in 3983  df-sn 4649  df-pr 4651  df-int 4971

This theorem is referenced by:  uniintsn  5009  intunsn  5011  op1stb  5491  op2ndb  6258  ssfii  9488  cf0  10320  cflecard  10322  uffix  23950  iotain  44386