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Theorem reapirr 8551
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 8579 instead. (Contributed by Jim Kingdon, 26-Jan-2020.)
Assertion
Ref Expression
reapirr (𝐴 ∈ ℝ → ¬ 𝐴 # 𝐴)

Proof of Theorem reapirr
StepHypRef Expression
1 ltnr 8051 . 2 (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴)
2 reapval 8550 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐴 # 𝐴 ↔ (𝐴 < 𝐴𝐴 < 𝐴)))
32anidms 397 . . 3 (𝐴 ∈ ℝ → (𝐴 # 𝐴 ↔ (𝐴 < 𝐴𝐴 < 𝐴)))
4 oridm 758 . . 3 ((𝐴 < 𝐴𝐴 < 𝐴) ↔ 𝐴 < 𝐴)
53, 4bitrdi 196 . 2 (𝐴 ∈ ℝ → (𝐴 # 𝐴𝐴 < 𝐴))
61, 5mtbird 674 1 (𝐴 ∈ ℝ → ¬ 𝐴 # 𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 105  wo 709  wcel 2159   class class class wbr 4017  cr 7827   < clt 8009   # creap 8548
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-sep 4135  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-setind 4550  ax-cnex 7919  ax-resscn 7920  ax-pre-ltirr 7940
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-nel 2455  df-ral 2472  df-rex 2473  df-rab 2476  df-v 2753  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-br 4018  df-opab 4079  df-xp 4646  df-pnf 8011  df-mnf 8012  df-ltxr 8014  df-reap 8549
This theorem is referenced by:  apirr  8579
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