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Metamath Proof Explorer

Theorem oaord1

Description: An ordinal is less than its sum with a nonzero ordinal. Theorem 18 of Suppes p. 209 and its converse. (Contributed by NM, 6-Dec-2004)

Ref Expression
Assertion oaord1 ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ∅ ∈ 𝐵 ↔ 𝐴 ∈ ( 𝐴 +o 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 0elon ⊢ ∅ ∈ On
2 oaord ⊢ ( ( ∅ ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On ) → ( ∅ ∈ 𝐵 ↔ ( 𝐴 +o ∅ ) ∈ ( 𝐴 +o 𝐵 ) ) )
3 1 2 mp3an1 ⊢ ( ( 𝐵 ∈ On ∧ 𝐴 ∈ On ) → ( ∅ ∈ 𝐵 ↔ ( 𝐴 +o ∅ ) ∈ ( 𝐴 +o 𝐵 ) ) )
4 oa0 ⊢ ( 𝐴 ∈ On → ( 𝐴 +o ∅ ) = 𝐴 )
5 4 adantl ⊢ ( ( 𝐵 ∈ On ∧ 𝐴 ∈ On ) → ( 𝐴 +o ∅ ) = 𝐴 )
6 5 eleq1d ⊢ ( ( 𝐵 ∈ On ∧ 𝐴 ∈ On ) → ( ( 𝐴 +o ∅ ) ∈ ( 𝐴 +o 𝐵 ) ↔ 𝐴 ∈ ( 𝐴 +o 𝐵 ) ) )
7 3 6 bitrd ⊢ ( ( 𝐵 ∈ On ∧ 𝐴 ∈ On ) → ( ∅ ∈ 𝐵 ↔ 𝐴 ∈ ( 𝐴 +o 𝐵 ) ) )
8 7 ancoms ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ∅ ∈ 𝐵 ↔ 𝐴 ∈ ( 𝐴 +o 𝐵 ) ) )