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Theorem ltso 6853
Description: 'Less than' is a strict ordering. (Contributed by NM, 19-Jan-1997.)
Assertion
Ref Expression
ltso  <  Or  RR

Proof of Theorem ltso
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltnr 6852 . . . . 5  RR  <
21adantl 262 . . . 4  RR  <
3 lttr 6849 . . . . 5  RR  RR  RR  <  <  <
43adantl 262 . . . 4  RR  RR  RR  <  <  <
52, 4ispod 4032 . . 3  <  Po  RR
65trud 1251 . 2  <  Po  RR
7 axltwlin 6844 . . 3  RR  RR  RR  <  <  <
87rgen3 2400 . 2  RR  RR  RR  <  <  <
9 df-iso 4025 . 2  < 
Or  RR  <  Po  RR  RR  RR  RR  <  <  <
106, 8, 9mpbir2an 848 1  <  Or  RR
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wa 97   wo 628   w3a 884   wtru 1243   wcel 1390  wral 2300   class class class wbr 3755    Po wpo 4022    Or wor 4023   RRcr 6670    < clt 6817
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-bnd 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 6734  ax-resscn 6735  ax-pre-ltirr 6755  ax-pre-ltwlin 6756  ax-pre-lttrn 6757
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-po 4024  df-iso 4025  df-xp 4294  df-pnf 6819  df-mnf 6820  df-ltxr 6822
This theorem is referenced by:  gtso  6854  ltnsym2  6865
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