Theorem intsn 3963

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 3962	. 2 ⊢ (A ∈ V → ∩{A} = A)
3	1, 2	ax-mp 5	1 ⊢ ∩{A} = A

Syntax hints:   = wceq 1642   ∈ wcel 1710  Vcvv 2860  {csn 3738  ∩cint 3927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-sn 3742  df-pr 3743  df-int 3928

This theorem is referenced by:  uniintsn  3964  intunsn  3966