Theorem ltexpi

Description: Ordering on positive integers in terms of existence of sum. (Contributed by NM, 15-Mar-1996) (Revised by Mario Carneiro, 14-Jun-2013) (New usage is discouraged.)

	Ref	Expression
Assertion	ltexpi	⊢ ( ( 𝐴 ∈ N ∧ 𝐵 ∈ N ) → ( 𝐴 <N 𝐵 ↔ ∃ 𝑥 ∈ N ( 𝐴 +N 𝑥 ) = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		pinn	⊢ ( 𝐴 ∈ N → 𝐴 ∈ ω )
2		pinn	⊢ ( 𝐵 ∈ N → 𝐵 ∈ ω )
3		nnaordex	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 ∈ 𝐵 ↔ ∃ 𝑥 ∈ ω ( ∅ ∈ 𝑥 ∧ ( 𝐴 +o 𝑥 ) = 𝐵 ) ) )
4	1 2 3	syl2an	⊢ ( ( 𝐴 ∈ N ∧ 𝐵 ∈ N ) → ( 𝐴 ∈ 𝐵 ↔ ∃ 𝑥 ∈ ω ( ∅ ∈ 𝑥 ∧ ( 𝐴 +o 𝑥 ) = 𝐵 ) ) )
5		ltpiord	⊢ ( ( 𝐴 ∈ N ∧ 𝐵 ∈ N ) → ( 𝐴 <N 𝐵 ↔ 𝐴 ∈ 𝐵 ) )
6		addpiord	⊢ ( ( 𝐴 ∈ N ∧ 𝑥 ∈ N ) → ( 𝐴 +N 𝑥 ) = ( 𝐴 +o 𝑥 ) )
7	6	eqeq1d	⊢ ( ( 𝐴 ∈ N ∧ 𝑥 ∈ N ) → ( ( 𝐴 +N 𝑥 ) = 𝐵 ↔ ( 𝐴 +o 𝑥 ) = 𝐵 ) )
8	7	pm5.32da	⊢ ( 𝐴 ∈ N → ( ( 𝑥 ∈ N ∧ ( 𝐴 +N 𝑥 ) = 𝐵 ) ↔ ( 𝑥 ∈ N ∧ ( 𝐴 +o 𝑥 ) = 𝐵 ) ) )
9		elni2	⊢ ( 𝑥 ∈ N ↔ ( 𝑥 ∈ ω ∧ ∅ ∈ 𝑥 ) )
10	9	anbi1i	⊢ ( ( 𝑥 ∈ N ∧ ( 𝐴 +o 𝑥 ) = 𝐵 ) ↔ ( ( 𝑥 ∈ ω ∧ ∅ ∈ 𝑥 ) ∧ ( 𝐴 +o 𝑥 ) = 𝐵 ) )
11		anass	⊢ ( ( ( 𝑥 ∈ ω ∧ ∅ ∈ 𝑥 ) ∧ ( 𝐴 +o 𝑥 ) = 𝐵 ) ↔ ( 𝑥 ∈ ω ∧ ( ∅ ∈ 𝑥 ∧ ( 𝐴 +o 𝑥 ) = 𝐵 ) ) )
12	10 11	bitri	⊢ ( ( 𝑥 ∈ N ∧ ( 𝐴 +o 𝑥 ) = 𝐵 ) ↔ ( 𝑥 ∈ ω ∧ ( ∅ ∈ 𝑥 ∧ ( 𝐴 +o 𝑥 ) = 𝐵 ) ) )
13	8 12	bitrdi	⊢ ( 𝐴 ∈ N → ( ( 𝑥 ∈ N ∧ ( 𝐴 +N 𝑥 ) = 𝐵 ) ↔ ( 𝑥 ∈ ω ∧ ( ∅ ∈ 𝑥 ∧ ( 𝐴 +o 𝑥 ) = 𝐵 ) ) ) )
14	13	rexbidv2	⊢ ( 𝐴 ∈ N → ( ∃ 𝑥 ∈ N ( 𝐴 +N 𝑥 ) = 𝐵 ↔ ∃ 𝑥 ∈ ω ( ∅ ∈ 𝑥 ∧ ( 𝐴 +o 𝑥 ) = 𝐵 ) ) )
15	14	adantr	⊢ ( ( 𝐴 ∈ N ∧ 𝐵 ∈ N ) → ( ∃ 𝑥 ∈ N ( 𝐴 +N 𝑥 ) = 𝐵 ↔ ∃ 𝑥 ∈ ω ( ∅ ∈ 𝑥 ∧ ( 𝐴 +o 𝑥 ) = 𝐵 ) ) )
16	4 5 15	3bitr4d	⊢ ( ( 𝐴 ∈ N ∧ 𝐵 ∈ N ) → ( 𝐴 <N 𝐵 ↔ ∃ 𝑥 ∈ N ( 𝐴 +N 𝑥 ) = 𝐵 ) )