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Theorem ltlen 9108
Description: 'Less than' expressed in terms of 'less than or equal to'. (Contributed by NM, 27-Oct-1999.)
Assertion
Ref Expression
ltlen  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  ( A  <_  B  /\  B  =/=  A ) ) )

Proof of Theorem ltlen
StepHypRef Expression
1 ltle 9096 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  ->  A  <_  B )
)
2 ltneOLD 9104 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  B  =/=  A )
323expia 1155 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  ->  B  =/=  A ) )
41, 3jcad 520 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  ->  ( A  <_  B  /\  B  =/=  A
) ) )
5 leloe 9094 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  <->  ( A  <  B  \/  A  =  B )
) )
6 ax-1 5 . . . . 5  |-  ( A  <  B  ->  ( B  =/=  A  ->  A  <  B ) )
7 df-ne 2552 . . . . . 6  |-  ( B  =/=  A  <->  -.  B  =  A )
8 pm2.24 103 . . . . . . 7  |-  ( B  =  A  ->  ( -.  B  =  A  ->  A  <  B ) )
98eqcoms 2390 . . . . . 6  |-  ( A  =  B  ->  ( -.  B  =  A  ->  A  <  B ) )
107, 9syl5bi 209 . . . . 5  |-  ( A  =  B  ->  ( B  =/=  A  ->  A  <  B ) )
116, 10jaoi 369 . . . 4  |-  ( ( A  <  B  \/  A  =  B )  ->  ( B  =/=  A  ->  A  <  B ) )
125, 11syl6bi 220 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  ->  ( B  =/=  A  ->  A  <  B ) ) )
1312imp3a 421 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  <_  B  /\  B  =/=  A
)  ->  A  <  B ) )
144, 13impbid 184 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  ( A  <_  B  /\  B  =/=  A ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2550   class class class wbr 4153   RRcr 8922    < clt 9053    <_ cle 9054
This theorem is referenced by:  ltleni  9122  ltlend  9150  rpneg  10573  fzm1  11057  elfznelfzob  11120  hashsdom  11582  cnpart  11972  ang180lem2  20519  mumullem2  20830  lgsneg  20970  lgsdilem  20973  lgsdirprm  20980  unitdivcld  24103  axlowdimlem16  25610
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-resscn 8980  ax-pre-lttri 8997  ax-pre-lttrn 8998
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-po 4444  df-so 4445  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059
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