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Theorem reapirr 7796
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 7824 instead. (Contributed by Jim Kingdon, 26-Jan-2020.)
Assertion
Ref Expression
reapirr  |-  ( A  e.  RR  ->  -.  A #  A )

Proof of Theorem reapirr
StepHypRef Expression
1 ltnr 7307 . 2  |-  ( A  e.  RR  ->  -.  A  <  A )
2 reapval 7795 . . . 4  |-  ( ( A  e.  RR  /\  A  e.  RR )  ->  ( A #  A  <->  ( A  < 
A  \/  A  < 
A ) ) )
32anidms 389 . . 3  |-  ( A  e.  RR  ->  ( A #  A 
<->  ( A  <  A  \/  A  <  A ) ) )
4 oridm 707 . . 3  |-  ( ( A  <  A  \/  A  <  A )  <->  A  <  A )
53, 4syl6bb 194 . 2  |-  ( A  e.  RR  ->  ( A #  A 
<->  A  <  A ) )
61, 5mtbird 631 1  |-  ( A  e.  RR  ->  -.  A #  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 103    \/ wo 662    e. wcel 1434   class class class wbr 3805   RRcr 7094    < clt 7267   # creap 7793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-cnex 7181  ax-resscn 7182  ax-pre-ltirr 7202
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2612  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-xp 4397  df-pnf 7269  df-mnf 7270  df-ltxr 7272  df-reap 7794
This theorem is referenced by:  apirr  7824
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