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Theorem axlttri 9081
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 8998 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 8998 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <RR  B  <->  -.  ( A  =  B  \/  B  <RR  A ) ) )
2 ltxrlt 9080 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  A 
<RR  B ) )
3 ltxrlt 9080 . . . . 5  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( B  <  A  <->  B 
<RR  A ) )
43ancoms 440 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( B  <  A  <->  B 
<RR  A ) )
54orbi2d 683 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  =  B  \/  B  < 
A )  <->  ( A  =  B  \/  B  <RR  A ) ) )
65notbid 286 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -.  ( A  =  B  \/  B  <  A )  <->  -.  ( A  =  B  \/  B  <RR  A ) ) )
71, 2, 63bitr4d 277 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 1649    e. wcel 1717   class class class wbr 4154   RRcr 8923    <RR cltrr 8928    < clt 9054
This theorem is referenced by:  ltso  9090  leloe  9095  ltnsym  9106  ltadd2  9111  lttrid  9144  ltord1  9486  recgt0  9787  recgt0ii  9849  arch  10151  xrlttri  10665  subgmulg  14886  cosord  20302  logdivlt  20384
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-resscn 8981  ax-pre-lttri 8998
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-pnf 9056  df-mnf 9057  df-ltxr 9059
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