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 e. RR /\ B e. RR /\ C e. RR ) -> ( ( A  A 

Detailed syntax breakdown

Step Hyp Ref Expression
0 cA
 |-  A
1 cr
 |-  RR
2 0 1 wcel
 |-  A e. RR
3 cB
 |-  B
4 3 1 wcel
 |-  B e. RR
5 cC
 |-  C
6 5 1 wcel
 |-  C e. RR
7 2 4 6 w3a
 |-  ( A e. RR /\ B e. RR /\ C e. RR )
8 cltrr
 |-  
9 0 3 8 wbr
 |-  A 
10 3 5 8 wbr
 |-  B 
11 9 10 wa
 |-  ( A 
12 0 5 8 wbr
 |-  A 
13 11 12 wi
 |-  ( ( A  A 
14 7 13 wi
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A  A