Metamath Proof Explorer


Axiom ax-pre-lttrn

Description: Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, justified by Theorem axpre-lttrn . Note: The more general version for extended reals is axlttrn . Normally new proofs would use lttr . (New usage is discouraged.) (Contributed by NM, 13-Oct-2005)

Ref Expression
Assertion ax-pre-lttrn A B C A < B B < C A < C

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 cC class C
6 5 1 wcel wff C
7 2 4 6 w3a wff A B C
8 cltrr class <
9 0 3 8 wbr wff A < B
10 3 5 8 wbr wff B < C
11 9 10 wa wff A < B B < C
12 0 5 8 wbr wff A < C
13 11 12 wi wff A < B B < C A < C
14 7 13 wi wff A B C A < B B < C A < C