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Theorem xrltnlet 5482
Description: 'Less than' expressed in terms of 'less than or equal to', for extended reals.
Assertion
Ref Expression
xrltnlet |- ((A e. RR* /\ B e. RR*) -> (A < B <-> -. B <_ A))
Proof of Theorem xrltnlet
StepHypRef Expression
1 xrlenltt 5481 . . 3 |- ((B e. RR* /\ A e. RR*) -> (B <_ A <-> -. A < B))
21con2bid 525 . 2 |- ((B e. RR* /\ A e. RR*) -> (A < B <-> -. B <_ A))
32ancoms 436 1 |- ((A e. RR* /\ B e. RR*) -> (A < B <-> -. B <_ A))
Colors of variables: wff set class
Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   e. wcel 956   class class class wbr 2614   <_ cle 5275  RR*cxr 5465   < clt 5466
This theorem is referenced by:  xrletrit 5542  ioo0t 6313  cdrci 10417
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-opab 2662  df-xp 3179  df-cnv 3181  df-le 5471
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