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Theorem dfle2 10497
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 8905 . 2  |-  Rel  <_
2 ltrelxr 8902 . . . 4  |-  <  C_  ( RR*  X.  RR* )
3 f1oi 5527 . . . . 5  |-  (  _I  |`  RR* ) : RR* -1-1-onto-> RR*
4 f1of 5488 . . . . 5  |-  ( (  _I  |`  RR* ) :
RR*
-1-1-onto-> RR* 
->  (  _I  |`  RR* ) : RR* --> RR* )
5 fssxp 5416 . . . . 5  |-  ( (  _I  |`  RR* ) :
RR* --> RR*  ->  (  _I  |` 
RR* )  C_  ( RR*  X.  RR* ) )
63, 4, 5mp2b 9 . . . 4  |-  (  _I  |`  RR* )  C_  ( RR*  X.  RR* )
72, 6unssi 3363 . . 3  |-  (  < 
u.  (  _I  |`  RR* )
)  C_  ( RR*  X. 
RR* )
8 relxp 4810 . . 3  |-  Rel  ( RR*  X.  RR* )
9 relss 4791 . . 3  |-  ( (  <  u.  (  _I  |`  RR* ) )  C_  ( RR*  X.  RR* )  ->  ( Rel  ( RR*  X. 
RR* )  ->  Rel  (  <  u.  (  _I  |`  RR* ) ) ) )
107, 8, 9mp2 17 . 2  |-  Rel  (  <  u.  (  _I  |`  RR* )
)
11 lerelxr 8904 . . . 4  |-  <_  C_  ( RR*  X.  RR* )
1211brel 4753 . . 3  |-  ( x  <_  y  ->  (
x  e.  RR*  /\  y  e.  RR* ) )
137brel 4753 . . 3  |-  ( x (  <  u.  (  _I  |`  RR* ) ) y  ->  ( x  e. 
RR*  /\  y  e.  RR* ) )
14 xrleloe 10494 . . . . 5  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
x  <_  y  <->  ( x  <  y  \/  x  =  y ) ) )
15 resieq 4981 . . . . . 6  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
x (  _I  |`  RR* )
y  <->  x  =  y
) )
1615orbi2d 682 . . . . 5  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
( x  <  y  \/  x (  _I  |`  RR* )
y )  <->  ( x  <  y  \/  x  =  y ) ) )
1714, 16bitr4d 247 . . . 4  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
x  <_  y  <->  ( x  <  y  \/  x (  _I  |`  RR* ) y ) ) )
18 brun 4085 . . . 4  |-  ( x (  <  u.  (  _I  |`  RR* ) ) y  <-> 
( x  <  y  \/  x (  _I  |`  RR* )
y ) )
1917, 18syl6bbr 254 . . 3  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
x  <_  y  <->  x (  <  u.  (  _I  |`  RR* )
) y ) )
2012, 13, 19pm5.21nii 342 . 2  |-  ( x  <_  y  <->  x (  <  u.  (  _I  |`  RR* )
) y )
211, 10, 20eqbrriv 4798 1  |-  <_  =  (  <  u.  (  _I  |`  RR* ) )
Colors of variables: wff set class
Syntax hints:    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696    u. cun 3163    C_ wss 3165   class class class wbr 4039    _I cid 4320    X. cxp 4703    |` cres 4707   Rel wrel 4710   -->wf 5267   -1-1-onto->wf1o 5270   RR*cxr 8882    < clt 8883    <_ cle 8884
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-pre-lttri 8827  ax-pre-lttrn 8828
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889
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