Metamath Proof Explorer


Theorem xrlttrd

Description: Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015)

Ref Expression
Hypotheses xrlttrd.1 ( 𝜑𝐴 ∈ ℝ* )
xrlttrd.2 ( 𝜑𝐵 ∈ ℝ* )
xrlttrd.3 ( 𝜑𝐶 ∈ ℝ* )
xrlttrd.4 ( 𝜑𝐴 < 𝐵 )
xrlttrd.5 ( 𝜑𝐵 < 𝐶 )
Assertion xrlttrd ( 𝜑𝐴 < 𝐶 )

Proof

Step Hyp Ref Expression
1 xrlttrd.1 ( 𝜑𝐴 ∈ ℝ* )
2 xrlttrd.2 ( 𝜑𝐵 ∈ ℝ* )
3 xrlttrd.3 ( 𝜑𝐶 ∈ ℝ* )
4 xrlttrd.4 ( 𝜑𝐴 < 𝐵 )
5 xrlttrd.5 ( 𝜑𝐵 < 𝐶 )
6 xrlttr ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ* ) → ( ( 𝐴 < 𝐵𝐵 < 𝐶 ) → 𝐴 < 𝐶 ) )
7 1 2 3 6 syl3anc ( 𝜑 → ( ( 𝐴 < 𝐵𝐵 < 𝐶 ) → 𝐴 < 𝐶 ) )
8 4 5 7 mp2and ( 𝜑𝐴 < 𝐶 )