Theorem intunsn 3966

Description: Theorem joining a singleton to an intersection. (Contributed by NM, 29-Sep-2002.)

Hypothesis

Ref	Expression
intunsn.1	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
intunsn	⊢ ∩ (𝐴 ∪ {𝐵}) = (∩ 𝐴 ∩ 𝐵)

Proof of Theorem intunsn

Step	Hyp	Ref	Expression
1		intun 3959	. 2 ⊢ ∩ (𝐴 ∪ {𝐵}) = (∩ 𝐴 ∩ ∩ {𝐵})
2		intunsn.1	. . . 4 ⊢ 𝐵 ∈ V
3	2	intsn 3963	. . 3 ⊢ ∩ {𝐵} = 𝐵
4	3	ineq2i 3405	. 2 ⊢ (∩ 𝐴 ∩ ∩ {𝐵}) = (∩ 𝐴 ∩ 𝐵)
5	1, 4	eqtri 2252	1 ⊢ ∩ (𝐴 ∪ {𝐵}) = (∩ 𝐴 ∩ 𝐵)

Syntax hints:   = wceq 1397   ∈ wcel 2202  Vcvv 2802   ∪ cun 3198   ∩ cin 3199  {csn 3669  ∩ cint 3928

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-v 2804  df-un 3204  df-in 3206  df-sn 3675  df-pr 3676  df-int 3929

This theorem is referenced by:  fiintim  7123