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Theorem ltlen 9167
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 9155 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  ->  A  <_  B )
)
2 ltneOLD 9163 . . . 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 9153 . . . 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 2600 . . . . . 6  |-  ( B  =/=  A  <->  -.  B  =  A )
8 pm2.24 103 . . . . . . 7  |-  ( B  =  A  ->  ( -.  B  =  A  ->  A  <  B ) )
98eqcoms 2438 . . . . . 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 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204   RRcr 8981    < clt 9112    <_ cle 9113
This theorem is referenced by:  ltleni  9183  ltlend  9210  rpneg  10633  fzm1  11119  elfznelfzob  11185  hashsdom  11647  cnpart  12037  ang180lem2  20644  mumullem2  20955  lgsneg  21095  lgsdilem  21098  lgsdirprm  21105  unitdivcld  24291  axlowdimlem16  25888  itg2addnclem  26246
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-pre-lttri 9056  ax-pre-lttrn 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118
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