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Axiom ax-pre-lttri 8827
Description: Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, justified by theorem axpre-lttri 8803. Note: The more general version for extended reals is axlttri 8910. Normally new proofs would use xrlttri 10489. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
Assertion
Ref Expression
ax-pre-lttri  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <RR  B  <->  -.  ( A  =  B  \/  B  <RR  A ) ) )

Detailed syntax breakdown of Axiom ax-pre-lttri
StepHypRef Expression
1 cA . . . 4  class  A
2 cr 8752 . . . 4  class  RR
31, 2wcel 1696 . . 3  wff  A  e.  RR
4 cB . . . 4  class  B
54, 2wcel 1696 . . 3  wff  B  e.  RR
63, 5wa 358 . 2  wff  ( A  e.  RR  /\  B  e.  RR )
7 cltrr 8757 . . . 4  class  <RR
81, 4, 7wbr 4039 . . 3  wff  A  <RR  B
91, 4wceq 1632 . . . . 5  wff  A  =  B
104, 1, 7wbr 4039 . . . . 5  wff  B  <RR  A
119, 10wo 357 . . . 4  wff  ( A  =  B  \/  B  <RR  A )
1211wn 3 . . 3  wff  -.  ( A  =  B  \/  B  <RR  A )
138, 12wb 176 . 2  wff  ( A 
<RR  B  <->  -.  ( A  =  B  \/  B  <RR  A ) )
146, 13wi 4 1  wff  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <RR  B  <->  -.  ( A  =  B  \/  B  <RR  A ) ) )
Colors of variables: wff set class
This axiom is referenced by:  axlttri  8910
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