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Theorem reapti 8538
Description: Real apartness is tight. Beyond the development of apartness itself, proofs should use apti 8581. (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 8036 . . . . 5 (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴)
21adantr 276 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ¬ 𝐴 < 𝐴)
3 oridm 757 . . . . . 6 ((𝐴 < 𝐴𝐴 < 𝐴) ↔ 𝐴 < 𝐴)
4 breq2 4009 . . . . . . 7 (𝐴 = 𝐵 → (𝐴 < 𝐴𝐴 < 𝐵))
5 breq1 4008 . . . . . . 7 (𝐴 = 𝐵 → (𝐴 < 𝐴𝐵 < 𝐴))
64, 5orbi12d 793 . . . . . 6 (𝐴 = 𝐵 → ((𝐴 < 𝐴𝐴 < 𝐴) ↔ (𝐴 < 𝐵𝐵 < 𝐴)))
73, 6bitr3id 194 . . . . 5 (𝐴 = 𝐵 → (𝐴 < 𝐴 ↔ (𝐴 < 𝐵𝐵 < 𝐴)))
87notbid 667 . . . 4 (𝐴 = 𝐵 → (¬ 𝐴 < 𝐴 ↔ ¬ (𝐴 < 𝐵𝐵 < 𝐴)))
92, 8syl5ibcom 155 . . 3 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 → ¬ (𝐴 < 𝐵𝐵 < 𝐴)))
10 reapval 8535 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 # 𝐵 ↔ (𝐴 < 𝐵𝐵 < 𝐴)))
1110notbid 667 . . 3 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ 𝐴 # 𝐵 ↔ ¬ (𝐴 < 𝐵𝐵 < 𝐴)))
129, 11sylibrd 169 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 → ¬ 𝐴 # 𝐵))
13 axapti 8030 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ (𝐴 < 𝐵𝐵 < 𝐴)) → 𝐴 = 𝐵)
14133expia 1205 . . 3 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ (𝐴 < 𝐵𝐵 < 𝐴) → 𝐴 = 𝐵))
1511, 14sylbid 150 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ 𝐴 # 𝐵𝐴 = 𝐵))
1612, 15impbid 129 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ ¬ 𝐴 # 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 708   = wceq 1353  wcel 2148   class class class wbr 4005  cr 7812   < clt 7994   # creap 8533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-pre-ltirr 7925  ax-pre-apti 7928
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-xp 4634  df-pnf 7996  df-mnf 7997  df-ltxr 7999  df-reap 8534
This theorem is referenced by:  rimul  8544  apreap  8546  apti  8581
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