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Theorem lttri3 6855
Description: Tightness of real apartness. (Contributed by NM, 5-May-1999.)
Assertion
Ref Expression
lttri3 ((A B ℝ) → (A = B ↔ (¬ A < B ¬ B < A)))

Proof of Theorem lttri3
StepHypRef Expression
1 ltnr 6852 . . . . 5 (A ℝ → ¬ A < A)
2 breq2 3759 . . . . . 6 (A = B → (A < AA < B))
32notbid 591 . . . . 5 (A = B → (¬ A < A ↔ ¬ A < B))
41, 3syl5ibcom 144 . . . 4 (A ℝ → (A = B → ¬ A < B))
5 breq1 3758 . . . . . 6 (A = B → (A < AB < A))
65notbid 591 . . . . 5 (A = B → (¬ A < A ↔ ¬ B < A))
71, 6syl5ibcom 144 . . . 4 (A ℝ → (A = B → ¬ B < A))
84, 7jcad 291 . . 3 (A ℝ → (A = B → (¬ A < B ¬ B < A)))
98adantr 261 . 2 ((A B ℝ) → (A = B → (¬ A < B ¬ B < A)))
10 ioran 668 . . 3 (¬ (A < B B < A) ↔ (¬ A < B ¬ B < A))
11 axapti 6847 . . . 4 ((A B ¬ (A < B B < A)) → A = B)
12113expia 1105 . . 3 ((A B ℝ) → (¬ (A < B B < A) → A = B))
1310, 12syl5bir 142 . 2 ((A B ℝ) → ((¬ A < B ¬ B < A) → A = B))
149, 13impbid 120 1 ((A B ℝ) → (A = B ↔ (¬ A < B ¬ B < A)))
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  cr 6670   < clt 6817
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 6734  ax-resscn 6735  ax-pre-ltirr 6755  ax-pre-apti 6758
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 6819  df-mnf 6820  df-ltxr 6822
This theorem is referenced by:  letri3  6856  lttri3i  6872  lttri3d  6889  inelr  7328  xrlttri3  8448
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