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Theorem lttri2 9157
Description: Consequence of trichotomy. (Contributed by NM, 9-Oct-1999.)
Assertion
Ref Expression
lttri2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  =/=  B  <->  ( A  <  B  \/  B  <  A ) ) )

Proof of Theorem lttri2
StepHypRef Expression
1 ltso 9156 . . . 4  |-  <  Or  RR
2 sotrieq 4530 . . . 4  |-  ( (  <  Or  RR  /\  ( A  e.  RR  /\  B  e.  RR ) )  ->  ( A  =  B  <->  -.  ( A  <  B  \/  B  < 
A ) ) )
31, 2mpan 652 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  =  B  <->  -.  ( A  <  B  \/  B  <  A ) ) )
43bicomd 193 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -.  ( A  <  B  \/  B  <  A )  <->  A  =  B ) )
54necon1abid 2657 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    = wceq 1652    e. wcel 1725    =/= wne 2599   class class class wbr 4212    Or wor 4502   RRcr 8989    < clt 9120
This theorem is referenced by:  ne0gt0  9178  lttri2i  9187  lttri2d  9212  dvne0  19895  atanlogsub  20756  itg2addnclem2  26257  nn0prpw  26326  nerabdioph  26869  rpnnen3  27103  fmul01lt1lem1  27690  cshwssizelem4  28284
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047  ax-pre-lttri 9064  ax-pre-lttrn 9065
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-ltxr 9125
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