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Theorem dfle2 10732
Description: Alternative definition of 'less than or equal to' in terms of 'less than'. (Contributed by Mario Carneiro, 6-Nov-2015.)
Assertion
Ref Expression
dfle2  |-  <_  =  (  <  u.  (  _I  |`  RR* ) )

Proof of Theorem dfle2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lerel 9134 . 2  |-  Rel  <_
2 ltrelxr 9131 . . . 4  |-  <  C_  ( RR*  X.  RR* )
3 f1oi 5705 . . . . 5  |-  (  _I  |`  RR* ) : RR* -1-1-onto-> RR*
4 f1of 5666 . . . . 5  |-  ( (  _I  |`  RR* ) :
RR*
-1-1-onto-> RR* 
->  (  _I  |`  RR* ) : RR* --> RR* )
5 fssxp 5594 . . . . 5  |-  ( (  _I  |`  RR* ) :
RR* --> RR*  ->  (  _I  |` 
RR* )  C_  ( RR*  X.  RR* ) )
63, 4, 5mp2b 10 . . . 4  |-  (  _I  |`  RR* )  C_  ( RR*  X.  RR* )
72, 6unssi 3514 . . 3  |-  (  < 
u.  (  _I  |`  RR* )
)  C_  ( RR*  X. 
RR* )
8 relxp 4975 . . 3  |-  Rel  ( RR*  X.  RR* )
9 relss 4955 . . 3  |-  ( (  <  u.  (  _I  |`  RR* ) )  C_  ( RR*  X.  RR* )  ->  ( Rel  ( RR*  X. 
RR* )  ->  Rel  (  <  u.  (  _I  |`  RR* ) ) ) )
107, 8, 9mp2 9 . 2  |-  Rel  (  <  u.  (  _I  |`  RR* )
)
11 lerelxr 9133 . . . 4  |-  <_  C_  ( RR*  X.  RR* )
1211brel 4918 . . 3  |-  ( x  <_  y  ->  (
x  e.  RR*  /\  y  e.  RR* ) )
137brel 4918 . . 3  |-  ( x (  <  u.  (  _I  |`  RR* ) ) y  ->  ( x  e. 
RR*  /\  y  e.  RR* ) )
14 xrleloe 10729 . . . . 5  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
x  <_  y  <->  ( x  <  y  \/  x  =  y ) ) )
15 resieq 5148 . . . . . 6  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
x (  _I  |`  RR* )
y  <->  x  =  y
) )
1615orbi2d 683 . . . . 5  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
( x  <  y  \/  x (  _I  |`  RR* )
y )  <->  ( x  <  y  \/  x  =  y ) ) )
1714, 16bitr4d 248 . . . 4  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
x  <_  y  <->  ( x  <  y  \/  x (  _I  |`  RR* ) y ) ) )
18 brun 4250 . . . 4  |-  ( x (  <  u.  (  _I  |`  RR* ) ) y  <-> 
( x  <  y  \/  x (  _I  |`  RR* )
y ) )
1917, 18syl6bbr 255 . . 3  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
x  <_  y  <->  x (  <  u.  (  _I  |`  RR* )
) y ) )
2012, 13, 19pm5.21nii 343 . 2  |-  ( x  <_  y  <->  x (  <  u.  (  _I  |`  RR* )
) y )
211, 10, 20eqbrriv 4963 1  |-  <_  =  (  <  u.  (  _I  |`  RR* ) )
Colors of variables: wff set class
Syntax hints:    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    u. cun 3310    C_ wss 3312   class class class wbr 4204    _I cid 4485    X. cxp 4868    |` cres 4872   Rel wrel 4875   -->wf 5442   -1-1-onto->wf1o 5445   RR*cxr 9111    < clt 9112    <_ cle 9113
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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