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Metamath Proof Explorer

Theorem onsseli

Description: Subset is equivalent to membership or equality for ordinal numbers. (Contributed by NM, 15-Sep-1995)

Ref Expression
Hypotheses on.1 ⊢ 𝐴 ∈ On
on.2 ⊢ 𝐵 ∈ On
Assertion onsseli ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ) )

Proof

Step Hyp Ref Expression
1 on.1 ⊢ 𝐴 ∈ On
2 on.2 ⊢ 𝐵 ∈ On
3 onsseleq ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ) ) )
4 1 2 3 mp2an ⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ) )