Database
ZF (ZERMELO-FRAENKEL) SET THEORY
ZF Set Theory - start with the Axiom of Extensionality
The intersection of a class
elrint2
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Indexed union and intersection
Metamath Proof Explorer
Ascii
Unicode
Theorem
elrint2
Description:
Membership in a restricted intersection.
(Contributed by
Stefan O'Rear
, 3-Apr-2015)
Ref
Expression
Assertion
elrint2
⊢
X
∈
A
→
X
∈
A
∩
⋂
B
↔
∀
y
∈
B
X
∈
y
Proof
Step
Hyp
Ref
Expression
1
elrint
⊢
X
∈
A
∩
⋂
B
↔
X
∈
A
∧
∀
y
∈
B
X
∈
y
2
1
baib
⊢
X
∈
A
→
X
∈
A
∩
⋂
B
↔
∀
y
∈
B
X
∈
y