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Theorem nltled 8107
Description: 'Not less than ' implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
ltd.1  |-  ( ph  ->  A  e.  RR )
ltd.2  |-  ( ph  ->  B  e.  RR )
nltled.1  |-  ( ph  ->  -.  B  <  A
)
Assertion
Ref Expression
nltled  |-  ( ph  ->  A  <_  B )

Proof of Theorem nltled
StepHypRef Expression
1 nltled.1 . 2  |-  ( ph  ->  -.  B  <  A
)
2 ltd.1 . . 3  |-  ( ph  ->  A  e.  RR )
3 ltd.2 . . 3  |-  ( ph  ->  B  e.  RR )
42, 3lenltd 8104 . 2  |-  ( ph  ->  ( A  <_  B  <->  -.  B  <  A ) )
51, 4mpbird 167 1  |-  ( ph  ->  A  <_  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 2160   class class class wbr 4018   RRcr 7839    < clt 8021    <_ cle 8022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4650  df-cnv 4652  df-xr 8025  df-le 8027
This theorem is referenced by:  ltntri  8114  suprubex  8937  infregelbex  9627  cvgratz  11571  zsupcl  11979  zssinfcl  11980  infssuzledc  11982  dvdslegcd  11996  pw2dvdseulemle  12198  dedekindeulemuub  14547  dedekindeulemlu  14551  suplociccex  14555  dedekindicclemuub  14556  dedekindicclemlu  14560  ivthinclemlopn  14566  ivthinclemuopn  14568  refeq  15230
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