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Metamath Proof Explorer

Theorem onsseleq

Description: Relationship between subset and membership of an ordinal number. (Contributed by NM, 15-Sep-1995)

Ref Expression
Assertion onsseleq ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 eloni ⊢ ( 𝐴 ∈ On → Ord 𝐴 )
2 eloni ⊢ ( 𝐵 ∈ On → Ord 𝐵 )
3 ordsseleq ⊢ ( ( Ord 𝐴 ∧ Ord 𝐵 ) → ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ) ) )
4 1 2 3 syl2an ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ) ) )