Metamath Proof Explorer


Axiom ax-pre-lttri

Description: Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, justified by Theorem axpre-lttri . Note: The more general version for extended reals is axlttri . Normally new proofs would use xrlttri . (New usage is discouraged.) (Contributed by NM, 13-Oct-2005)

Ref Expression
Assertion ax-pre-lttri A B A < B ¬ A = B B < A

Detailed syntax breakdown

Step Hyp Ref Expression
0 cA class A
1 cr class
2 0 1 wcel wff A
3 cB class B
4 3 1 wcel wff B
5 2 4 wa wff A B
6 cltrr class <
7 0 3 6 wbr wff A < B
8 0 3 wceq wff A = B
9 3 0 6 wbr wff B < A
10 8 9 wo wff A = B B < A
11 10 wn wff ¬ A = B B < A
12 7 11 wb wff A < B ¬ A = B B < A
13 5 12 wi wff A B A < B ¬ A = B B < A