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Axiom ax-pre-apti 7056
Description: Apartness of reals is tight. Axiom for real and complex numbers, justified by theorem axpre-apti 7016. (Contributed by Jim Kingdon, 29-Jan-2020.)
Assertion
Ref Expression
ax-pre-apti  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  -.  ( A  <RR  B  \/  B  <RR  A ) )  ->  A  =  B )

Detailed syntax breakdown of Axiom ax-pre-apti
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 cltrr 6950 . . . . . 6  class  <RR
71, 4, 6wbr 3791 . . . . 5  wff  A  <RR  B
84, 1, 6wbr 3791 . . . . 5  wff  B  <RR  A
97, 8wo 639 . . . 4  wff  ( A 
<RR  B  \/  B  <RR  A )
109wn 3 . . 3  wff  -.  ( A  <RR  B  \/  B  <RR  A )
113, 5, 10w3a 896 . 2  wff  ( A  e.  RR  /\  B  e.  RR  /\  -.  ( A  <RR  B  \/  B  <RR  A ) )
121, 4wceq 1259 . 2  wff  A  =  B
1311, 12wi 4 1  wff  ( ( A  e.  RR  /\  B  e.  RR  /\  -.  ( A  <RR  B  \/  B  <RR  A ) )  ->  A  =  B )
Colors of variables: wff set class
This axiom is referenced by:  axapti  7148
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