Theorem naddss2

Description: Ordinal less-than-or-equal is not affected by natural addition. (Contributed by Scott Fenton, 9-Sep-2024)

	Ref	Expression
Assertion	naddss2	⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐴 ⊆ 𝐵 ↔ ( 𝐶 +no 𝐴 ) ⊆ ( 𝐶 +no 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		naddss1	⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 +no 𝐶 ) ⊆ ( 𝐵 +no 𝐶 ) ) )
2		naddcom	⊢ ( ( 𝐴 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐴 +no 𝐶 ) = ( 𝐶 +no 𝐴 ) )
3	2	3adant2	⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐴 +no 𝐶 ) = ( 𝐶 +no 𝐴 ) )
4		naddcom	⊢ ( ( 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐵 +no 𝐶 ) = ( 𝐶 +no 𝐵 ) )
5	4	3adant1	⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐵 +no 𝐶 ) = ( 𝐶 +no 𝐵 ) )
6	3 5	sseq12d	⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( ( 𝐴 +no 𝐶 ) ⊆ ( 𝐵 +no 𝐶 ) ↔ ( 𝐶 +no 𝐴 ) ⊆ ( 𝐶 +no 𝐵 ) ) )
7	1 6	bitrd	⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐴 ⊆ 𝐵 ↔ ( 𝐶 +no 𝐴 ) ⊆ ( 𝐶 +no 𝐵 ) ) )