Database
ZF (ZERMELO-FRAENKEL) SET THEORY
ZF Set Theory - add the Axiom of Power Sets
Ordinals
ordsseleq
Metamath Proof Explorer
Description: For ordinal classes, inclusion is equivalent to membership or equality.
(Contributed by NM , 25-Nov-1995) (Proof shortened by Andrew Salmon , 25-Jul-2011)
Ref
Expression
Assertion
ordsseleq
⊢ Ord ⁡ A ∧ Ord ⁡ B → A ⊆ B ↔ A ∈ B ∨ A = B
Proof
Step
Hyp
Ref
Expression
1
ordelpss
⊢ Ord ⁡ A ∧ Ord ⁡ B → A ∈ B ↔ A ⊂ B
2
1
orbi1d
⊢ Ord ⁡ A ∧ Ord ⁡ B → A ∈ B ∨ A = B ↔ A ⊂ B ∨ A = B
3
sspss
⊢ A ⊆ B ↔ A ⊂ B ∨ A = B
4
2 3
syl6rbbr
⊢ Ord ⁡ A ∧ Ord ⁡ B → A ⊆ B ↔ A ∈ B ∨ A = B