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Theorem reapti 7363
 Description: Real apartness is tight. Beyond the development of apartness itself, proofs should use apti 7406. (Contributed by Jim Kingdon, 30-Jan-2020.) (New usage is discouraged.)
Assertion
Ref Expression
reapti ((A B ℝ) → (A = B ↔ ¬ A # B))

Proof of Theorem reapti
StepHypRef Expression
1 ltnr 6892 . . . . 5 (A ℝ → ¬ A < A)
21adantr 261 . . . 4 ((A B ℝ) → ¬ A < A)
3 oridm 673 . . . . . 6 ((A < A A < A) ↔ A < A)
4 breq2 3759 . . . . . . 7 (A = B → (A < AA < B))
5 breq1 3758 . . . . . . 7 (A = B → (A < AB < A))
64, 5orbi12d 706 . . . . . 6 (A = B → ((A < A A < A) ↔ (A < B B < A)))
73, 6syl5bbr 183 . . . . 5 (A = B → (A < A ↔ (A < B B < A)))
87notbid 591 . . . 4 (A = B → (¬ A < A ↔ ¬ (A < B B < A)))
92, 8syl5ibcom 144 . . 3 ((A B ℝ) → (A = B → ¬ (A < B B < A)))
10 reapval 7360 . . . 4 ((A B ℝ) → (A # B ↔ (A < B B < A)))
1110notbid 591 . . 3 ((A B ℝ) → (¬ A # B ↔ ¬ (A < B B < A)))
129, 11sylibrd 158 . 2 ((A B ℝ) → (A = B → ¬ A # B))
13 axapti 6887 . . . 4 ((A B ¬ (A < B B < A)) → A = B)
14133expia 1105 . . 3 ((A B ℝ) → (¬ (A < B B < A) → A = B))
1511, 14sylbid 139 . 2 ((A B ℝ) → (¬ A # BA = B))
1612, 15impbid 120 1 ((A B ℝ) → (A = B ↔ ¬ A # B))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 628   = wceq 1242   ∈ wcel 1390   class class class wbr 3755  ℝcr 6710   < clt 6857   #ℝ creap 7358 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 6774  ax-resscn 6775  ax-pre-ltirr 6795  ax-pre-apti 6798 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 6859  df-mnf 6860  df-ltxr 6862  df-reap 7359 This theorem is referenced by:  rimul  7369  apreap  7371  apti  7406
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