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Axiom ax-pre-ltwlin 7054
Description: Real number less-than is weakly linear. Axiom for real and complex numbers, justified by theorem axpre-ltwlin 7014. (Contributed by Jim Kingdon, 12-Jan-2020.)
Assertion
Ref Expression
ax-pre-ltwlin  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <RR  B  ->  ( A  <RR  C  \/  C  <RR  B ) ) )

Detailed syntax breakdown of Axiom ax-pre-ltwlin
StepHypRef Expression
1 cA . . . 4  class  A
2 cr 6945 . . . 4  class  RR
31, 2wcel 1409 . . 3  wff  A  e.  RR
4 cB . . . 4  class  B
54, 2wcel 1409 . . 3  wff  B  e.  RR
6 cC . . . 4  class  C
76, 2wcel 1409 . . 3  wff  C  e.  RR
83, 5, 7w3a 896 . 2  wff  ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )
9 cltrr 6950 . . . 4  class  <RR
101, 4, 9wbr 3791 . . 3  wff  A  <RR  B
111, 6, 9wbr 3791 . . . 4  wff  A  <RR  C
126, 4, 9wbr 3791 . . . 4  wff  C  <RR  B
1311, 12wo 639 . . 3  wff  ( A 
<RR  C  \/  C  <RR  B )
1410, 13wi 4 . 2  wff  ( A 
<RR  B  ->  ( A  <RR  C  \/  C  <RR  B ) )
158, 14wi 4 1  wff  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <RR  B  ->  ( A  <RR  C  \/  C  <RR  B ) ) )
Colors of variables: wff set class
This axiom is referenced by:  axltwlin  7145
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