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Theorem axapti 6867
Description: Apartness of reals is tight. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-apti 6778 with ordering on the extended reals.) (Contributed by Jim Kingdon, 29-Jan-2020.)
Assertion
Ref Expression
axapti ((A B ¬ (A < B B < A)) → A = B)

Proof of Theorem axapti
StepHypRef Expression
1 ltxrlt 6862 . . . . 5 ((A B ℝ) → (A < BA < B))
2 ltxrlt 6862 . . . . . 6 ((B A ℝ) → (B < AB < A))
32ancoms 255 . . . . 5 ((A B ℝ) → (B < AB < A))
41, 3orbi12d 706 . . . 4 ((A B ℝ) → ((A < B B < A) ↔ (A < B B < A)))
54notbid 591 . . 3 ((A B ℝ) → (¬ (A < B B < A) ↔ ¬ (A < B B < A)))
6 ax-pre-apti 6778 . . . 4 ((A B ¬ (A < B B < A)) → A = B)
763expia 1105 . . 3 ((A B ℝ) → (¬ (A < B B < A) → A = B))
85, 7sylbid 139 . 2 ((A B ℝ) → (¬ (A < B B < A) → A = B))
983impia 1100 1 ((A B ¬ (A < B B < A)) → A = B)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wo 628   w3a 884   = wceq 1242   wcel 1390   class class class wbr 3755  cr 6690   < cltrr 6695   < clt 6837
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-cnex 6754  ax-resscn 6755  ax-pre-apti 6778
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-pnf 6839  df-mnf 6840  df-ltxr 6842
This theorem is referenced by:  lttri3  6875  reapti  7343
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