Database
ZF (ZERMELO-FRAENKEL) SET THEORY
ZF Set Theory - add the Axiom of Union
Finite sets
dmfi
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fundmfibi
Metamath Proof Explorer
Ascii
Unicode
Theorem
dmfi
Description:
The domain of a finite set is finite.
(Contributed by
Mario Carneiro
, 24-Sep-2013)
Ref
Expression
Assertion
dmfi
⊢
A
∈
Fin
→
dom
⁡
A
∈
Fin
Proof
Step
Hyp
Ref
Expression
1
fidomdm
⊢
A
∈
Fin
→
dom
⁡
A
≼
A
2
domfi
⊢
A
∈
Fin
∧
dom
⁡
A
≼
A
→
dom
⁡
A
∈
Fin
3
1
2
mpdan
⊢
A
∈
Fin
→
dom
⁡
A
∈
Fin