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Theorem dfle2 10481
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 8889 . 2  |-  Rel  <_
2 ltrelxr 8886 . . . 4  |-  <  C_  ( RR*  X.  RR* )
3 f1oi 5511 . . . . 5  |-  (  _I  |`  RR* ) : RR* -1-1-onto-> RR*
4 f1of 5472 . . . . 5  |-  ( (  _I  |`  RR* ) :
RR*
-1-1-onto-> RR* 
->  (  _I  |`  RR* ) : RR* --> RR* )
5 fssxp 5400 . . . . 5  |-  ( (  _I  |`  RR* ) :
RR* --> RR*  ->  (  _I  |` 
RR* )  C_  ( RR*  X.  RR* ) )
63, 4, 5mp2b 9 . . . 4  |-  (  _I  |`  RR* )  C_  ( RR*  X.  RR* )
72, 6unssi 3350 . . 3  |-  (  < 
u.  (  _I  |`  RR* )
)  C_  ( RR*  X. 
RR* )
8 relxp 4794 . . 3  |-  Rel  ( RR*  X.  RR* )
9 relss 4775 . . 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 8888 . . . 4  |-  <_  C_  ( RR*  X.  RR* )
1211brel 4737 . . 3  |-  ( x  <_  y  ->  (
x  e.  RR*  /\  y  e.  RR* ) )
137brel 4737 . . 3  |-  ( x (  <  u.  (  _I  |`  RR* ) ) y  ->  ( x  e. 
RR*  /\  y  e.  RR* ) )
14 xrleloe 10478 . . . . 5  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
x  <_  y  <->  ( x  <  y  \/  x  =  y ) ) )
15 resieq 4965 . . . . . 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 4069 . . . 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 4782 1  |-  <_  =  (  <  u.  (  _I  |`  RR* ) )
Colors of variables: wff set class
Syntax hints:    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    u. cun 3150    C_ wss 3152   class class class wbr 4023    _I cid 4304    X. cxp 4687    |` cres 4691   Rel wrel 4694   -->wf 5251   -1-1-onto->wf1o 5254   RR*cxr 8866    < clt 8867    <_ cle 8868
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-pre-lttri 8811  ax-pre-lttrn 8812
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873
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