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Theorem ltne 9103
Description: 'Less than' implies not equal. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 16-Sep-2015.)
Assertion
Ref Expression
ltne  |-  ( ( A  e.  RR  /\  A  <  B )  ->  B  =/=  A )

Proof of Theorem ltne
StepHypRef Expression
1 ltnr 9101 . . . 4  |-  ( A  e.  RR  ->  -.  A  <  A )
2 breq2 4157 . . . . 5  |-  ( B  =  A  ->  ( A  <  B  <->  A  <  A ) )
32notbid 286 . . . 4  |-  ( B  =  A  ->  ( -.  A  <  B  <->  -.  A  <  A ) )
41, 3syl5ibrcom 214 . . 3  |-  ( A  e.  RR  ->  ( B  =  A  ->  -.  A  <  B ) )
54necon2ad 2598 . 2  |-  ( A  e.  RR  ->  ( A  <  B  ->  B  =/=  A ) )
65imp 419 1  |-  ( ( A  e.  RR  /\  A  <  B )  ->  B  =/=  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2550   class class class wbr 4153   RRcr 8922    < clt 9053
This theorem is referenced by:  ltneOLD  9104  gtneii  9116  gtned  9140  gt0ne0  9425  lt0ne0  9426  gt0ne0d  9523  nvnencycllem  21478  staddi  23597  axlowdimlem16  25610  ftc1cnnc  25979  fdc  26140
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-ltxr 9058
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