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Metamath Proof Explorer

Theorem axlttrn

Description: Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, derived from ZF set theory. This restates ax-pre-lttrn with ordering on the extended reals. New proofs should use lttr instead for naming consistency. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005)

Ref Expression
Assertion axlttrn ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 < 𝐵 ∧ 𝐵 < 𝐶 ) → 𝐴 < 𝐶 ) )

Proof

Step Hyp Ref Expression
1 ax-pre-lttrn ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ 𝐶 ) → 𝐴 <ℝ 𝐶 ) )
2 ltxrlt ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 < 𝐵 ↔ 𝐴 <ℝ 𝐵 ) )
3 2 3adant3 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 < 𝐵 ↔ 𝐴 <ℝ 𝐵 ) )
4 ltxrlt ⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐵 < 𝐶 ↔ 𝐵 <ℝ 𝐶 ) )
5 4 3adant1 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐵 < 𝐶 ↔ 𝐵 <ℝ 𝐶 ) )
6 3 5 anbi12d ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 < 𝐵 ∧ 𝐵 < 𝐶 ) ↔ ( 𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ 𝐶 ) ) )
7 ltxrlt ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 < 𝐶 ↔ 𝐴 <ℝ 𝐶 ) )
8 7 3adant2 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 < 𝐶 ↔ 𝐴 <ℝ 𝐶 ) )
9 1 6 8 3imtr4d ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 < 𝐵 ∧ 𝐵 < 𝐶 ) → 𝐴 < 𝐶 ) )