Theorem intunsn 4917

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 4908	. 2 ⊢ ∩ (𝐴 ∪ {𝐵}) = (∩ 𝐴 ∩ ∩ {𝐵})
2		intunsn.1	. . . 4 ⊢ 𝐵 ∈ V
3	2	intsn 4914	. . 3 ⊢ ∩ {𝐵} = 𝐵
4	3	ineq2i 4140	. 2 ⊢ (∩ 𝐴 ∩ ∩ {𝐵}) = (∩ 𝐴 ∩ 𝐵)
5	1, 4	eqtri 2766	1 ⊢ ∩ (𝐴 ∪ {𝐵}) = (∩ 𝐴 ∩ 𝐵)

Syntax hints:   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ∪ cun 3881   ∩ cin 3882  {csn 4558  ∩ cint 4876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rab 3072  df-v 3424  df-un 3888  df-in 3890  df-sn 4559  df-pr 4561  df-int 4877

This theorem is referenced by:  fiint  9021  incexclem  15476  heibor1lem  35894