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

Theorem reapirr 8302
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 8330 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 7805 . 2  |-  ( A  e.  RR  ->  -.  A  <  A )
2 reapval 8301 . . . 4  |-  ( ( A  e.  RR  /\  A  e.  RR )  ->  ( A #  A  <->  ( A  < 
A  \/  A  < 
A ) ) )
32anidms 392 . . 3  |-  ( A  e.  RR  ->  ( A #  A 
<->  ( A  <  A  \/  A  <  A ) ) )
4 oridm 729 . . 3  |-  ( ( A  <  A  \/  A  <  A )  <->  A  <  A )
53, 4syl6bb 195 . 2  |-  ( A  e.  RR  ->  ( A #  A 
<->  A  <  A ) )
61, 5mtbird 645 1  |-  ( A  e.  RR  ->  -.  A #  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 104    \/ wo 680    e. wcel 1463   class class class wbr 3897   RRcr 7583    < clt 7764   # creap 8299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-cnex 7675  ax-resscn 7676  ax-pre-ltirr 7696
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-xp 4513  df-pnf 7766  df-mnf 7767  df-ltxr 7769  df-reap 8300
This theorem is referenced by:  apirr  8330
  Copyright terms: Public domain W3C validator