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Theorem axlttri 8891
Description: Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, derived from ZF set theory. (This restates ax-pre-lttri 8808 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.)
Assertion
Ref Expression
axlttri  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  -.  ( A  =  B  \/  B  <  A
) ) )

Proof of Theorem axlttri
StepHypRef Expression
1 ax-pre-lttri 8808 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <RR  B  <->  -.  ( A  =  B  \/  B  <RR  A ) ) )
2 ltxrlt 8890 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  A 
<RR  B ) )
3 ltxrlt 8890 . . . . 5  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( B  <  A  <->  B 
<RR  A ) )
43ancoms 441 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( B  <  A  <->  B 
<RR  A ) )
54orbi2d 684 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  =  B  \/  B  < 
A )  <->  ( A  =  B  \/  B  <RR  A ) ) )
65notbid 287 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -.  ( A  =  B  \/  B  <  A )  <->  -.  ( A  =  B  \/  B  <RR  A ) ) )
71, 2, 63bitr4d 278 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  -.  ( A  =  B  \/  B  <  A
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1625    e. wcel 1687   class class class wbr 4026   RRcr 8733    <RR cltrr 8738    < clt 8864
This theorem is referenced by:  ltso  8900  leloe  8905  ltnsym  8916  ltadd2  8921  lttrid  8954  ltord1  9296  recgt0  9597  recgt0ii  9659  arch  9959  xrlttri  10470  subgmulg  14631  cosord  19890  logdivlt  19968
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-13 1689  ax-14 1691  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267  ax-sep 4144  ax-nul 4152  ax-pow 4189  ax-pr 4215  ax-un 4513  ax-resscn 8791  ax-pre-lttri 8808
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1531  df-nf 1534  df-sb 1633  df-eu 2150  df-mo 2151  df-clab 2273  df-cleq 2279  df-clel 2282  df-nfc 2411  df-ne 2451  df-nel 2452  df-ral 2551  df-rex 2552  df-rab 2555  df-v 2793  df-sbc 2995  df-csb 3085  df-dif 3158  df-un 3160  df-in 3162  df-ss 3169  df-nul 3459  df-if 3569  df-pw 3630  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3831  df-br 4027  df-opab 4081  df-mpt 4082  df-id 4310  df-xp 4696  df-rel 4697  df-cnv 4698  df-co 4699  df-dm 4700  df-rn 4701  df-res 4702  df-ima 4703  df-fun 5225  df-fn 5226  df-f 5227  df-f1 5228  df-fo 5229  df-f1o 5230  df-fv 5231  df-er 6657  df-en 6861  df-dom 6862  df-sdom 6863  df-pnf 8866  df-mnf 8867  df-ltxr 8869
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