Theorem intunsn 4957

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 4950	. 2 ⊢ ∩ (𝐴 ∪ {𝐵}) = (∩ 𝐴 ∩ ∩ {𝐵})
2		intunsn.1	. . . 4 ⊢ 𝐵 ∈ V
3	2	intsn 4954	. . 3 ⊢ ∩ {𝐵} = 𝐵
4	3	ineq2i 4173	. 2 ⊢ (∩ 𝐴 ∩ ∩ {𝐵}) = (∩ 𝐴 ∩ 𝐵)
5	1, 4	eqtri 2789	1 ⊢ ∩ (𝐴 ∪ {𝐵}) = (∩ 𝐴 ∩ 𝐵)

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2146  Vcvv 3458   ∪ cun 3906   ∩ cin 3907  {csn 4594  ∩ cint 4917

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-un 3913  df-in 3915  df-sn 4595  df-pr 4597  df-int 4918

This theorem is used by:  fiint  9296  incexclem  15916  heibor1lem  38501