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Theorem reapti 7643
Description: Real apartness is tight. Beyond the development of apartness itself, proofs should use apti 7686. (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 7153 . . . . 5 (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴)
21adantr 265 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ¬ 𝐴 < 𝐴)
3 oridm 684 . . . . . 6 ((𝐴 < 𝐴𝐴 < 𝐴) ↔ 𝐴 < 𝐴)
4 breq2 3795 . . . . . . 7 (𝐴 = 𝐵 → (𝐴 < 𝐴𝐴 < 𝐵))
5 breq1 3794 . . . . . . 7 (𝐴 = 𝐵 → (𝐴 < 𝐴𝐵 < 𝐴))
64, 5orbi12d 717 . . . . . 6 (𝐴 = 𝐵 → ((𝐴 < 𝐴𝐴 < 𝐴) ↔ (𝐴 < 𝐵𝐵 < 𝐴)))
73, 6syl5bbr 187 . . . . 5 (𝐴 = 𝐵 → (𝐴 < 𝐴 ↔ (𝐴 < 𝐵𝐵 < 𝐴)))
87notbid 602 . . . 4 (𝐴 = 𝐵 → (¬ 𝐴 < 𝐴 ↔ ¬ (𝐴 < 𝐵𝐵 < 𝐴)))
92, 8syl5ibcom 148 . . 3 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 → ¬ (𝐴 < 𝐵𝐵 < 𝐴)))
10 reapval 7640 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 # 𝐵 ↔ (𝐴 < 𝐵𝐵 < 𝐴)))
1110notbid 602 . . 3 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ 𝐴 # 𝐵 ↔ ¬ (𝐴 < 𝐵𝐵 < 𝐴)))
129, 11sylibrd 162 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 → ¬ 𝐴 # 𝐵))
13 axapti 7148 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ (𝐴 < 𝐵𝐵 < 𝐴)) → 𝐴 = 𝐵)
14133expia 1117 . . 3 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ (𝐴 < 𝐵𝐵 < 𝐴) → 𝐴 = 𝐵))
1511, 14sylbid 143 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ 𝐴 # 𝐵𝐴 = 𝐵))
1612, 15impbid 124 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ ¬ 𝐴 # 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wb 102  wo 639   = wceq 1259  wcel 1409   class class class wbr 3791  cr 6945   < clt 7118   # creap 7638
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-cnex 7032  ax-resscn 7033  ax-pre-ltirr 7053  ax-pre-apti 7056
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-xp 4378  df-pnf 7120  df-mnf 7121  df-ltxr 7123  df-reap 7639
This theorem is referenced by:  rimul  7649  apreap  7651  apti  7686
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