Theorem intsn 3814

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 3813	. 2 ⊢ (𝐴 ∈ V → ∩ {𝐴} = 𝐴)
3	1, 2	ax-mp 5	1 ⊢ ∩ {𝐴} = 𝐴

Syntax hints:   = wceq 1332   ∈ wcel 1481  Vcvv 2689  {csn 3532  ∩ cint 3779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2691  df-un 3080  df-in 3082  df-sn 3538  df-pr 3539  df-int 3780

This theorem is referenced by:  uniintsnr  3815  intunsn  3817  op1stb  4407  op2ndb  5030  ssfii  6870