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Database ZF (ZERMELO-FRAENKEL) SET THEORY ZF Set Theory - add the Axiom of Power Sets Ordinals ordsseleq
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
sspss ⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵 ) )
2
ordelpss ⊢ ( ( Ord 𝐴 ∧ Ord 𝐵 ) → ( 𝐴 ∈ 𝐵 ↔ 𝐴 ⊊ 𝐵 ) )
3
2
orbi1d ⊢ ( ( Ord 𝐴 ∧ Ord 𝐵 ) → ( ( 𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ) ↔ ( 𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵 ) ) )
4
1 3
bitr4id ⊢ ( ( Ord 𝐴 ∧ Ord 𝐵 ) → ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ) ) )