Theorem intunsn 3961

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 3954	. 2 ⊢ ∩ (𝐴 ∪ {𝐵}) = (∩ 𝐴 ∩ ∩ {𝐵})
2		intunsn.1	. . . 4 ⊢ 𝐵 ∈ V
3	2	intsn 3958	. . 3 ⊢ ∩ {𝐵} = 𝐵
4	3	ineq2i 3402	. 2 ⊢ (∩ 𝐴 ∩ ∩ {𝐵}) = (∩ 𝐴 ∩ 𝐵)
5	1, 4	eqtri 2250	1 ⊢ ∩ (𝐴 ∪ {𝐵}) = (∩ 𝐴 ∩ 𝐵)

Syntax hints:   = wceq 1395   ∈ wcel 2200  Vcvv 2799   ∪ cun 3195   ∩ cin 3196  {csn 3666  ∩ cint 3923

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2801  df-un 3201  df-in 3203  df-sn 3672  df-pr 3673  df-int 3924

This theorem is referenced by:  fiintim  7101