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Theorem ltne 7842
Description: 'Less than' implies not equal. See also ltap 8388 which is the same but for apartness. (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 7834 . . . 4  |-  ( A  e.  RR  ->  -.  A  <  A )
2 breq2 3928 . . . . 5  |-  ( B  =  A  ->  ( A  <  B  <->  A  <  A ) )
32notbid 656 . . . 4  |-  ( B  =  A  ->  ( -.  A  <  B  <->  -.  A  <  A ) )
41, 3syl5ibrcom 156 . . 3  |-  ( A  e.  RR  ->  ( B  =  A  ->  -.  A  <  B ) )
54necon2ad 2363 . 2  |-  ( A  e.  RR  ->  ( A  <  B  ->  B  =/=  A ) )
65imp 123 1  |-  ( ( A  e.  RR  /\  A  <  B )  ->  B  =/=  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480    =/= wne 2306   class class class wbr 3924   RRcr 7612    < clt 7793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-pre-ltirr 7725
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-xp 4540  df-pnf 7795  df-mnf 7796  df-ltxr 7798
This theorem is referenced by:  gtneii  7852  ltnei  7860  gtned  7869  gt0ne0  8182  lt0ne0  8183  gt0ne0d  8267  nngt1ne1  8748  zdceq  9119  qdceq  10017  coprm  11811  phibndlem  11881
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