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Theorem leltne 9156
Description: 'Less than or equal to' implies 'less than' is not 'equals'. (Contributed by NM, 27-Jul-1999.)
Assertion
Ref Expression
leltne  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( A  <  B  <->  B  =/=  A ) )

Proof of Theorem leltne
StepHypRef Expression
1 lttri3 9150 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  =  B  <-> 
( -.  A  < 
B  /\  -.  B  <  A ) ) )
2 simpl 444 . . . . . . 7  |-  ( ( -.  A  <  B  /\  -.  B  <  A
)  ->  -.  A  <  B )
31, 2syl6bi 220 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  =  B  ->  -.  A  <  B ) )
43adantr 452 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  ( A  =  B  ->  -.  A  <  B ) )
5 leloe 9153 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  <->  ( A  <  B  \/  A  =  B )
) )
65biimpa 471 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  ( A  <  B  \/  A  =  B ) )
76ord 367 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  ( -.  A  <  B  ->  A  =  B ) )
84, 7impbid 184 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  ( A  =  B  <->  -.  A  <  B ) )
98necon2abid 2655 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  ( A  <  B  <->  A  =/=  B
) )
10 necom 2679 . . 3  |-  ( B  =/=  A  <->  A  =/=  B )
119, 10syl6bbr 255 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  ( A  <  B  <->  B  =/=  A
) )
12113impa 1148 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    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204   RRcr 8981    < clt 9112    <_ cle 9113
This theorem is referenced by:  leltned  9216  nngt1ne1  10019  gcdn0gt0  13014  isprm3  13080  iundisj2  19435  norm-i  22623  cnlnadjlem7  23568  iundisj2f  24022  iundisj2fi  24145  fmul01lt1lem1  27681  cshwssizelem2  28244
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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