Database
ZF (ZERMELO-FRAENKEL) SET THEORY
ZF Set Theory - start with the Axiom of Extensionality
Subclasses and subsets
eqimssi
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eqimss2i
Metamath Proof Explorer
Ascii
Structured
Theorem
eqimssi
Description:
Infer subclass relationship from equality.
(Contributed by
NM
, 6-Jan-2007)
Ref
Expression
Hypothesis
eqimssi.1
⊢
𝐴
=
𝐵
Assertion
eqimssi
⊢
𝐴
⊆
𝐵
Proof
Step
Hyp
Ref
Expression
1
eqimssi.1
⊢
𝐴
=
𝐵
2
ssid
⊢
𝐴
⊆
𝐴
3
2
1
sseqtri
⊢
𝐴
⊆
𝐵