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Metamath Proof Explorer

Theorem oaword2

Description: An ordinal is less than or equal to its sum with another. Theorem 21 of Suppes p. 209. Lemma 3.3 of Schloeder p. 7. (Contributed by NM, 7-Dec-2004)

Ref Expression
Assertion oaword2 ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → 𝐴 ⊆ ( 𝐵 +o 𝐴 ) )

Proof

Step Hyp Ref Expression
1 0ss ⊢ ∅ ⊆ 𝐵
2 0elon ⊢ ∅ ∈ On
3 oawordri ⊢ ( ( ∅ ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On ) → ( ∅ ⊆ 𝐵 → ( ∅ +o 𝐴 ) ⊆ ( 𝐵 +o 𝐴 ) ) )
4 2 3 mp3an1 ⊢ ( ( 𝐵 ∈ On ∧ 𝐴 ∈ On ) → ( ∅ ⊆ 𝐵 → ( ∅ +o 𝐴 ) ⊆ ( 𝐵 +o 𝐴 ) ) )
5 oa0r ⊢ ( 𝐴 ∈ On → ( ∅ +o 𝐴 ) = 𝐴 )
6 5 adantl ⊢ ( ( 𝐵 ∈ On ∧ 𝐴 ∈ On ) → ( ∅ +o 𝐴 ) = 𝐴 )
7 6 sseq1d ⊢ ( ( 𝐵 ∈ On ∧ 𝐴 ∈ On ) → ( ( ∅ +o 𝐴 ) ⊆ ( 𝐵 +o 𝐴 ) ↔ 𝐴 ⊆ ( 𝐵 +o 𝐴 ) ) )
8 4 7 sylibd ⊢ ( ( 𝐵 ∈ On ∧ 𝐴 ∈ On ) → ( ∅ ⊆ 𝐵 → 𝐴 ⊆ ( 𝐵 +o 𝐴 ) ) )
9 1 8 mpi ⊢ ( ( 𝐵 ∈ On ∧ 𝐴 ∈ On ) → 𝐴 ⊆ ( 𝐵 +o 𝐴 ) )
10 9 ancoms ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → 𝐴 ⊆ ( 𝐵 +o 𝐴 ) )