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 ) → ( ∅ ∈ 𝐵𝐴 ∈ ( 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 ) → ( ∅ ∈ 𝐵𝐴 ∈ ( 𝐴 +o 𝐵 ) ) )
8 7 ancoms ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ∅ ∈ 𝐵𝐴 ∈ ( 𝐴 +o 𝐵 ) ) )