Theorem nnaordr

Description: Ordering property of addition of natural numbers. (Contributed by NM, 9-Nov-2002)

	Ref	Expression
Assertion	nnaordr	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω ) → ( 𝐴 ∈ 𝐵 ↔ ( 𝐴 +o 𝐶 ) ∈ ( 𝐵 +o 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		nnaord	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω ) → ( 𝐴 ∈ 𝐵 ↔ ( 𝐶 +o 𝐴 ) ∈ ( 𝐶 +o 𝐵 ) ) )
2		nnacom	⊢ ( ( 𝐶 ∈ ω ∧ 𝐴 ∈ ω ) → ( 𝐶 +o 𝐴 ) = ( 𝐴 +o 𝐶 ) )
3	2	ancoms	⊢ ( ( 𝐴 ∈ ω ∧ 𝐶 ∈ ω ) → ( 𝐶 +o 𝐴 ) = ( 𝐴 +o 𝐶 ) )
4	3	3adant2	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω ) → ( 𝐶 +o 𝐴 ) = ( 𝐴 +o 𝐶 ) )
5		nnacom	⊢ ( ( 𝐶 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐶 +o 𝐵 ) = ( 𝐵 +o 𝐶 ) )
6	5	ancoms	⊢ ( ( 𝐵 ∈ ω ∧ 𝐶 ∈ ω ) → ( 𝐶 +o 𝐵 ) = ( 𝐵 +o 𝐶 ) )
7	6	3adant1	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω ) → ( 𝐶 +o 𝐵 ) = ( 𝐵 +o 𝐶 ) )
8	4 7	eleq12d	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω ) → ( ( 𝐶 +o 𝐴 ) ∈ ( 𝐶 +o 𝐵 ) ↔ ( 𝐴 +o 𝐶 ) ∈ ( 𝐵 +o 𝐶 ) ) )
9	1 8	bitrd	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω ) → ( 𝐴 ∈ 𝐵 ↔ ( 𝐴 +o 𝐶 ) ∈ ( 𝐵 +o 𝐶 ) ) )