ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  reapirr Structured version   GIF version

Theorem reapirr 7321
Description: Real apartness is irreflexive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). Beyond the development of # itself, proofs should use apirr 7349 instead. (Contributed by Jim Kingdon, 26-Jan-2020.)
Assertion
Ref Expression
reapirr (A ℝ → ¬ A # A)

Proof of Theorem reapirr
StepHypRef Expression
1 ltnr 6852 . 2 (A ℝ → ¬ A < A)
2 reapval 7320 . . . 4 ((A A ℝ) → (A # A ↔ (A < A A < A)))
32anidms 377 . . 3 (A ℝ → (A # A ↔ (A < A A < A)))
4 oridm 673 . . 3 ((A < A A < A) ↔ A < A)
53, 4syl6bb 185 . 2 (A ℝ → (A # AA < A))
61, 5mtbird 597 1 (A ℝ → ¬ A # A)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 98   wo 628   wcel 1390   class class class wbr 3755  cr 6670   < clt 6817   # creap 7318
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
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  df-reap 7319
This theorem is referenced by:  apirr  7349
  Copyright terms: Public domain W3C validator