ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ax-pre-lttrn Structured version   GIF version

Axiom ax-pre-lttrn 6757
Description: Ordering on reals is transitive. Axiom for real and complex numbers, justified by theorem axpre-lttrn 6728. (Contributed by NM, 13-Oct-2005.)
Assertion
Ref Expression
ax-pre-lttrn ((A B 𝐶 ℝ) → ((A < B B < 𝐶) → A < 𝐶))

Detailed syntax breakdown of Axiom ax-pre-lttrn
StepHypRef Expression
1 cA . . . 4 class A
2 cr 6670 . . . 4 class
31, 2wcel 1390 . . 3 wff A
4 cB . . . 4 class B
54, 2wcel 1390 . . 3 wff B
6 cC . . . 4 class 𝐶
76, 2wcel 1390 . . 3 wff 𝐶
83, 5, 7w3a 884 . 2 wff (A B 𝐶 ℝ)
9 cltrr 6675 . . . . 5 class <
101, 4, 9wbr 3755 . . . 4 wff A < B
114, 6, 9wbr 3755 . . . 4 wff B < 𝐶
1210, 11wa 97 . . 3 wff (A < B B < 𝐶)
131, 6, 9wbr 3755 . . 3 wff A < 𝐶
1412, 13wi 4 . 2 wff ((A < B B < 𝐶) → A < 𝐶)
158, 14wi 4 1 wff ((A B 𝐶 ℝ) → ((A < B B < 𝐶) → A < 𝐶))
Colors of variables: wff set class
This axiom is referenced by:  axlttrn  6845
  Copyright terms: Public domain W3C validator