Theorem intsn 3926

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 3925	. 2 ⊢ (𝐴 ∈ V → ∩ {𝐴} = 𝐴)
3	1, 2	ax-mp 5	1 ⊢ ∩ {𝐴} = 𝐴

Syntax hints:   = wceq 1373   ∈ wcel 2177  Vcvv 2773  {csn 3638  ∩ cint 3891

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-v 2775  df-un 3174  df-in 3176  df-sn 3644  df-pr 3645  df-int 3892

This theorem is referenced by:  uniintsnr  3927  intunsn  3929  op1stb  4533  op2ndb  5175  ssfii  7091