Database
ZF (ZERMELO-FRAENKEL) SET THEORY
ZF Set Theory - start with the Axiom of Extensionality
Transitive classes
tr0
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Metamath Proof Explorer
Ascii
Structured
Theorem
tr0
Description:
The empty set is transitive.
(Contributed by
NM
, 16-Sep-1993)
Ref
Expression
Assertion
tr0
⊢
Tr ∅
Proof
Step
Hyp
Ref
Expression
1
0ss
⊢
∅ ⊆ 𝒫 ∅
2
dftr4
⊢
( Tr ∅ ↔ ∅ ⊆ 𝒫 ∅ )
3
1
2
mpbir
⊢
Tr ∅