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 3455	. 2 ⊢ (∩A ∩ ∩{B}) = (∩A ∩ B)
5	1, 4	eqtri 2373	1 ⊢ ∩(A ∪ {B}) = (∩A ∩ B)

Syntax hints:   = wceq 1642   ∈ wcel 1710  Vcvv 2860   ∪ cun 3208   ∩ cin 3209  {csn 3738  ∩cint 3927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-sn 3742  df-pr 3743  df-int 3928

This theorem is referenced by: (None)