This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) )