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Theorem reapti 8353
Description: Real apartness is tight. Beyond the development of apartness itself, proofs should use apti 8396. (Contributed by Jim Kingdon, 30-Jan-2020.) (New usage is discouraged.)
Assertion
Ref Expression
reapti ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ ¬ 𝐴 # 𝐵))

Proof of Theorem reapti
StepHypRef Expression
1 ltnr 7853 . . . . 5 (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴)
21adantr 274 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ¬ 𝐴 < 𝐴)
3 oridm 746 . . . . . 6 ((𝐴 < 𝐴𝐴 < 𝐴) ↔ 𝐴 < 𝐴)
4 breq2 3933 . . . . . . 7 (𝐴 = 𝐵 → (𝐴 < 𝐴𝐴 < 𝐵))
5 breq1 3932 . . . . . . 7 (𝐴 = 𝐵 → (𝐴 < 𝐴𝐵 < 𝐴))
64, 5orbi12d 782 . . . . . 6 (𝐴 = 𝐵 → ((𝐴 < 𝐴𝐴 < 𝐴) ↔ (𝐴 < 𝐵𝐵 < 𝐴)))
73, 6bitr3id 193 . . . . 5 (𝐴 = 𝐵 → (𝐴 < 𝐴 ↔ (𝐴 < 𝐵𝐵 < 𝐴)))
87notbid 656 . . . 4 (𝐴 = 𝐵 → (¬ 𝐴 < 𝐴 ↔ ¬ (𝐴 < 𝐵𝐵 < 𝐴)))
92, 8syl5ibcom 154 . . 3 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 → ¬ (𝐴 < 𝐵𝐵 < 𝐴)))
10 reapval 8350 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 # 𝐵 ↔ (𝐴 < 𝐵𝐵 < 𝐴)))
1110notbid 656 . . 3 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ 𝐴 # 𝐵 ↔ ¬ (𝐴 < 𝐵𝐵 < 𝐴)))
129, 11sylibrd 168 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 → ¬ 𝐴 # 𝐵))
13 axapti 7847 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ (𝐴 < 𝐵𝐵 < 𝐴)) → 𝐴 = 𝐵)
14133expia 1183 . . 3 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ (𝐴 < 𝐵𝐵 < 𝐴) → 𝐴 = 𝐵))
1511, 14sylbid 149 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ 𝐴 # 𝐵𝐴 = 𝐵))
1612, 15impbid 128 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ ¬ 𝐴 # 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 697   = wceq 1331  wcel 1480   class class class wbr 3929  cr 7631   < clt 7812   # creap 8348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7723  ax-resscn 7724  ax-pre-ltirr 7744  ax-pre-apti 7747
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-xp 4545  df-pnf 7814  df-mnf 7815  df-ltxr 7817  df-reap 8349
This theorem is referenced by:  rimul  8359  apreap  8361  apti  8396
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