Theorem intunsn 1993

Description: Theorem joining a singleton to an intersection.

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 1989	. 2 ⊢ ∩(A ∪ {B}) = (∩A ∩ ∩{B})
2		intunsn.1	. . . 4 ⊢ B ∈ V
3	2	intsn 1991	. . 3 ⊢ ∩{B} = B
4	3	ineq2i 1642	. 2 ⊢ (∩A ∩ ∩{B}) = (∩A ∩ B)
5	1, 4	eqtr 1119	1 ⊢ ∩(A ∪ {B}) = (∩A ∩ B)

Syntax hints:   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ∪ cun 1485   ∩ cin 1486  {csn 1808  ∩cint 1965

This theorem is referenced by:  fiint 3445

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-un 1490  df-in 1491  df-sn 1811  df-pr 1812  df-int 1966

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