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Theorem reapirr 7361
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 7389 instead. (Contributed by Jim Kingdon, 26-Jan-2020.)
Assertion
Ref Expression
reapirr  RR #

Proof of Theorem reapirr
StepHypRef Expression
1 ltnr 6892 . 2  RR  <
2 reapval 7360 . . . 4  RR  RR #  <  <
32anidms 377 . . 3  RR #  <  <
4 oridm 673 . . 3  <  <  <
53, 4syl6bb 185 . 2  RR #  <
61, 5mtbird 597 1  RR #
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wb 98   wo 628   wcel 1390   class class class wbr 3755   RRcr 6710    < clt 6857   # 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-bndl 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
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:  apirr  7389
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