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Axiom ax-pre-apti 6778
 Description: Apartness of reals is tight. Axiom for real and complex numbers, justified by theorem axpre-apti 6749. (Contributed by Jim Kingdon, 29-Jan-2020.)
Assertion
Ref Expression
ax-pre-apti ((A B ¬ (A < B B < A)) → A = B)

Detailed syntax breakdown of Axiom ax-pre-apti
StepHypRef Expression
1 cA . . . 4 class A
2 cr 6690 . . . 4 class
31, 2wcel 1390 . . 3 wff A
4 cB . . . 4 class B
54, 2wcel 1390 . . 3 wff B
6 cltrr 6695 . . . . . 6 class <
71, 4, 6wbr 3755 . . . . 5 wff A < B
84, 1, 6wbr 3755 . . . . 5 wff B < A
97, 8wo 628 . . . 4 wff (A < B B < A)
109wn 3 . . 3 wff ¬ (A < B B < A)
113, 5, 10w3a 884 . 2 wff (A B ¬ (A < B B < A))
121, 4wceq 1242 . 2 wff A = B
1311, 12wi 4 1 wff ((A B ¬ (A < B B < A)) → A = B)
 Colors of variables: wff set class This axiom is referenced by:  axapti  6867
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