Database
ZF (ZERMELO-FRAENKEL) SET THEORY
ZF Set Theory - start with the Axiom of Extensionality
Subclasses and subsets
pssssd
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pssned
Metamath Proof Explorer
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Theorem
pssssd
Description:
Deduce subclass from proper subclass.
(Contributed by
NM
, 29-Feb-1996)
Ref
Expression
Hypothesis
pssssd.1
⊢
φ
→
A
⊂
B
Assertion
pssssd
⊢
φ
→
A
⊆
B
Proof
Step
Hyp
Ref
Expression
1
pssssd.1
⊢
φ
→
A
⊂
B
2
pssss
⊢
A
⊂
B
→
A
⊆
B
3
1
2
syl
⊢
φ
→
A
⊆
B