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Theorem reapirr 8115
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 8143 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 7623 . 2  |-  ( A  e.  RR  ->  -.  A  <  A )
2 reapval 8114 . . . 4  |-  ( ( A  e.  RR  /\  A  e.  RR )  ->  ( A #  A  <->  ( A  < 
A  \/  A  < 
A ) ) )
32anidms 390 . . 3  |-  ( A  e.  RR  ->  ( A #  A 
<->  ( A  <  A  \/  A  <  A ) ) )
4 oridm 710 . . 3  |-  ( ( A  <  A  \/  A  <  A )  <->  A  <  A )
53, 4syl6bb 195 . 2  |-  ( A  e.  RR  ->  ( A #  A 
<->  A  <  A ) )
61, 5mtbird 634 1  |-  ( A  e.  RR  ->  -.  A #  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 104    \/ wo 665    e. wcel 1439   class class class wbr 3851   RRcr 7410    < clt 7583   # creap 8112
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-cnex 7497  ax-resscn 7498  ax-pre-ltirr 7518
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-rab 2369  df-v 2622  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-xp 4458  df-pnf 7585  df-mnf 7586  df-ltxr 7588  df-reap 8113
This theorem is referenced by:  apirr  8143
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