Theorem intunsn 5011

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 5004	. 2 ⊢ ∩ (𝐴 ∪ {𝐵}) = (∩ 𝐴 ∩ ∩ {𝐵})
2		intunsn.1	. . . 4 ⊢ 𝐵 ∈ V
3	2	intsn 5008	. . 3 ⊢ ∩ {𝐵} = 𝐵
4	3	ineq2i 4238	. 2 ⊢ (∩ 𝐴 ∩ ∩ {𝐵}) = (∩ 𝐴 ∩ 𝐵)
5	1, 4	eqtri 2768	1 ⊢ ∩ (𝐴 ∪ {𝐵}) = (∩ 𝐴 ∩ 𝐵)

Syntax hints:   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ∪ cun 3974   ∩ cin 3975  {csn 4648  ∩ cint 4970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-un 3981  df-in 3983  df-sn 4649  df-pr 4651  df-int 4971

This theorem is referenced by:  fiint  9394  fiintOLD  9395  incexclem  15884  heibor1lem  37769