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Theorem leltne 8911
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 8905 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  =  B  <-> 
( -.  A  < 
B  /\  -.  B  <  A ) ) )
2 simpl 443 . . . . . . 7  |-  ( ( -.  A  <  B  /\  -.  B  <  A
)  ->  -.  A  <  B )
31, 2syl6bi 219 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  =  B  ->  -.  A  <  B ) )
43adantr 451 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  ( A  =  B  ->  -.  A  <  B ) )
5 leloe 8908 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  <->  ( A  <  B  \/  A  =  B )
) )
65biimpa 470 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  ( A  <  B  \/  A  =  B ) )
76ord 366 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  ( -.  A  <  B  ->  A  =  B ) )
84, 7impbid 183 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  ( A  =  B  <->  -.  A  <  B ) )
98necon2abid 2503 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  ( A  <  B  <->  A  =/=  B
) )
10 necom 2527 . . 3  |-  ( B  =/=  A  <->  A  =/=  B )
119, 10syl6bbr 254 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  ( A  <  B  <->  B  =/=  A
) )
12113impa 1146 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 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   RRcr 8736    < clt 8867    <_ cle 8868
This theorem is referenced by:  leltned  8970  nngt1ne1  9773  gcdn0gt0  12701  isprm3  12767  iundisj2  18906  norm-i  21708  cnlnadjlem7  22653  iundisj2fi  23364  iundisj2f  23366  fmul01lt1lem1  27714
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-pre-lttri 8811  ax-pre-lttrn 8812
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873
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