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Theorem xrltlen 10731
Description: 'Less than' expressed in terms of 'less than or equal to'. (Contributed by Mario Carneiro, 6-Nov-2015.)
Assertion
Ref Expression
xrltlen  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  ( A  <_  B  /\  B  =/= 
A ) ) )

Proof of Theorem xrltlen
StepHypRef Expression
1 xrlttri 10724 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  -.  ( A  =  B  \/  B  <  A ) ) )
2 ioran 477 . . . 4  |-  ( -.  ( A  =  B  \/  B  <  A
)  <->  ( -.  A  =  B  /\  -.  B  <  A ) )
3 ancom 438 . . . 4  |-  ( ( -.  A  =  B  /\  -.  B  < 
A )  <->  ( -.  B  <  A  /\  -.  A  =  B )
)
42, 3bitri 241 . . 3  |-  ( -.  ( A  =  B  \/  B  <  A
)  <->  ( -.  B  <  A  /\  -.  A  =  B ) )
51, 4syl6bb 253 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  ( -.  B  <  A  /\  -.  A  =  B )
) )
6 xrlenlt 9135 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  <->  -.  B  <  A ) )
7 necom 2679 . . . . 5  |-  ( B  =/=  A  <->  A  =/=  B )
8 df-ne 2600 . . . . 5  |-  ( A  =/=  B  <->  -.  A  =  B )
97, 8bitri 241 . . . 4  |-  ( B  =/=  A  <->  -.  A  =  B )
109a1i 11 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( B  =/=  A  <->  -.  A  =  B ) )
116, 10anbi12d 692 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  <_  B  /\  B  =/=  A
)  <->  ( -.  B  <  A  /\  -.  A  =  B ) ) )
125, 11bitr4d 248 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 2598   class class class wbr 4204   RR*cxr 9111    < clt 9112    <_ cle 9113
This theorem is referenced by:  dflt2  10733  hashgt0  11654
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-pre-lttri 9056  ax-pre-lttrn 9057
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118
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