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Theorem lttri2i 9079
Description: Consequence of trichotomy. (Contributed by NM, 19-Jan-1997.)
Hypotheses
Ref Expression
lt.1  |-  A  e.  RR
lt.2  |-  B  e.  RR
Assertion
Ref Expression
lttri2i  |-  ( A  =/=  B  <->  ( A  <  B  \/  B  < 
A ) )

Proof of Theorem lttri2i
StepHypRef Expression
1 lt.1 . 2  |-  A  e.  RR
2 lt.2 . 2  |-  B  e.  RR
3 lttri2 9051 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  =/=  B  <->  ( A  <  B  \/  B  <  A ) ) )
41, 2, 3mp2an 653 1  |-  ( A  =/=  B  <->  ( A  <  B  \/  B  < 
A ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    \/ wo 357    e. wcel 1715    =/= wne 2529   class class class wbr 4125   RRcr 8883    < clt 9014
This theorem is referenced by:  nn0opthi  11450  dvdslelem  12781  divalglem6  12805
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-resscn 8941  ax-pre-lttri 8958  ax-pre-lttrn 8959
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-po 4417  df-so 4418  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-pnf 9016  df-mnf 9017  df-ltxr 9019
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