Theorem intsn 4951

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 4950	. 2 ⊢ (𝐴 ∈ V → ∩ {𝐴} = 𝐴)
3	1, 2	ax-mp 5	1 ⊢ ∩ {𝐴} = 𝐴

Syntax hints:   = wceq 1540   ∈ wcel 2109  Vcvv 3450  {csn 4592  ∩ cint 4913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-v 3452  df-un 3922  df-in 3924  df-sn 4593  df-pr 4595  df-int 4914

This theorem is referenced by:  uniintsn  4952  intunsn  4954  op1stb  5434  op2ndb  6203  ssfii  9377  cf0  10211  cflecard  10213  uffix  23815  iotain  44413