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Axiom ax-pre-lttrn 8812
Description: Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, justified by theorem axpre-lttrn 8788. Note: The more general version for extended reals is axlttrn 8895. Normally new proofs would use lttr 8899. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
Assertion
Ref Expression
ax-pre-lttrn  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  <RR  B  /\  B  <RR  C )  ->  A  <RR  C ) )

Detailed syntax breakdown of Axiom ax-pre-lttrn
StepHypRef Expression
1 cA . . . 4  class  A
2 cr 8736 . . . 4  class  RR
31, 2wcel 1684 . . 3  wff  A  e.  RR
4 cB . . . 4  class  B
54, 2wcel 1684 . . 3  wff  B  e.  RR
6 cC . . . 4  class  C
76, 2wcel 1684 . . 3  wff  C  e.  RR
83, 5, 7w3a 934 . 2  wff  ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )
9 cltrr 8741 . . . . 5  class  <RR
101, 4, 9wbr 4023 . . . 4  wff  A  <RR  B
114, 6, 9wbr 4023 . . . 4  wff  B  <RR  C
1210, 11wa 358 . . 3  wff  ( A 
<RR  B  /\  B  <RR  C )
131, 6, 9wbr 4023 . . 3  wff  A  <RR  C
1412, 13wi 4 . 2  wff  ( ( A  <RR  B  /\  B  <RR  C )  ->  A  <RR  C )
158, 14wi 4 1  wff  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  <RR  B  /\  B  <RR  C )  ->  A  <RR  C ) )
Colors of variables: wff set class
This axiom is referenced by:  axlttrn  8895
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