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Metamath Proof Explorer

Theorem oaword1

Description: An ordinal is less than or equal to its sum with another. Part of Exercise 5 of TakeutiZaring p. 62. Lemma 3.2 of Schloeder p. 7. (For the other part see oaord1 .) (Contributed by NM, 6-Dec-2004)

Ref Expression
Assertion oaword1 ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → 𝐴 ⊆ ( 𝐴 +o 𝐵 ) )

Proof

Step Hyp Ref Expression
1 oa0 ⊢ ( 𝐴 ∈ On → ( 𝐴 +o ∅ ) = 𝐴 )
2 1 adantr ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 +o ∅ ) = 𝐴 )
3 0ss ⊢ ∅ ⊆ 𝐵
4 0elon ⊢ ∅ ∈ On
5 oaword ⊢ ( ( ∅ ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On ) → ( ∅ ⊆ 𝐵 ↔ ( 𝐴 +o ∅ ) ⊆ ( 𝐴 +o 𝐵 ) ) )
6 5 3com13 ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ ∅ ∈ On ) → ( ∅ ⊆ 𝐵 ↔ ( 𝐴 +o ∅ ) ⊆ ( 𝐴 +o 𝐵 ) ) )
7 4 6 mp3an3 ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ∅ ⊆ 𝐵 ↔ ( 𝐴 +o ∅ ) ⊆ ( 𝐴 +o 𝐵 ) ) )
8 3 7 mpbii ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 +o ∅ ) ⊆ ( 𝐴 +o 𝐵 ) )
9 2 8 eqsstrrd ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → 𝐴 ⊆ ( 𝐴 +o 𝐵 ) )