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Theorem ltlen 9753
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 9740 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  ->  A  <_  B )
)
2 ltneOLD 9749 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  B  =/=  A )
323expia 1233 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  ->  B  =/=  A ) )
41, 3jcad 542 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  ->  ( A  <_  B  /\  B  =/=  A
) ) )
5 leloe 9738 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  <->  ( A  <  B  \/  A  =  B )
) )
6 ax-1 6 . . . . 5  |-  ( A  <  B  ->  ( B  =/=  A  ->  A  <  B ) )
7 df-ne 2643 . . . . . 6  |-  ( B  =/=  A  <->  -.  B  =  A )
8 pm2.24 112 . . . . . . 7  |-  ( B  =  A  ->  ( -.  B  =  A  ->  A  <  B ) )
98eqcoms 2479 . . . . . 6  |-  ( A  =  B  ->  ( -.  B  =  A  ->  A  <  B ) )
107, 9syl5bi 225 . . . . 5  |-  ( A  =  B  ->  ( B  =/=  A  ->  A  <  B ) )
116, 10jaoi 386 . . . 4  |-  ( ( A  <  B  \/  A  =  B )  ->  ( B  =/=  A  ->  A  <  B ) )
125, 11syl6bi 236 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  ->  ( B  =/=  A  ->  A  <  B ) ) )
1312impd 438 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  <_  B  /\  B  =/=  A
)  ->  A  <  B ) )
144, 13impbid 195 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  ( A  <_  B  /\  B  =/=  A ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   class class class wbr 4395   RRcr 9556    < clt 9693    <_ cle 9694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-resscn 9614  ax-pre-lttri 9631  ax-pre-lttrn 9632
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-po 4760  df-so 4761  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699
This theorem is referenced by:  ltleni  9770  ltlend  9797  nn0lt2  11023  rpneg  11355  fzofzim  11990  elfznelfzob  12046  hashsdom  12598  cnpart  13380  chfacfisf  19955  chfacfisfcpmat  19956  ang180lem2  23818  mumullem2  24186  lgsneg  24326  lgsdilem  24329  lgsdirprm  24336  axlowdimlem16  25066  unitdivcld  28781  poimirlem24  32028  itg2addnclem  32057  fzopredsuc  38865  iccpartiltu  38881  icceuelpartlem  38894  difmodm1lt  40833
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