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Theorem dfle2 11356
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 9640 . 2  |-  Rel  <_
2 ltrelxr 9637 . . . 4  |-  <  C_  ( RR*  X.  RR* )
3 f1oi 5833 . . . . 5  |-  (  _I  |`  RR* ) : RR* -1-1-onto-> RR*
4 f1of 5798 . . . . 5  |-  ( (  _I  |`  RR* ) :
RR*
-1-1-onto-> RR* 
->  (  _I  |`  RR* ) : RR* --> RR* )
5 fssxp 5725 . . . . 5  |-  ( (  _I  |`  RR* ) :
RR* --> RR*  ->  (  _I  |` 
RR* )  C_  ( RR*  X.  RR* ) )
63, 4, 5mp2b 10 . . . 4  |-  (  _I  |`  RR* )  C_  ( RR*  X.  RR* )
72, 6unssi 3665 . . 3  |-  (  < 
u.  (  _I  |`  RR* )
)  C_  ( RR*  X. 
RR* )
8 relxp 5098 . . 3  |-  Rel  ( RR*  X.  RR* )
9 relss 5078 . . 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 9639 . . . 4  |-  <_  C_  ( RR*  X.  RR* )
1211brel 5037 . . 3  |-  ( x  <_  y  ->  (
x  e.  RR*  /\  y  e.  RR* ) )
137brel 5037 . . 3  |-  ( x (  <  u.  (  _I  |`  RR* ) ) y  ->  ( x  e. 
RR*  /\  y  e.  RR* ) )
14 xrleloe 11353 . . . . 5  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
x  <_  y  <->  ( x  <  y  \/  x  =  y ) ) )
15 resieq 5272 . . . . . 6  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
x (  _I  |`  RR* )
y  <->  x  =  y
) )
1615orbi2d 699 . . . . 5  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
( x  <  y  \/  x (  _I  |`  RR* )
y )  <->  ( x  <  y  \/  x  =  y ) ) )
1714, 16bitr4d 256 . . . 4  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
x  <_  y  <->  ( x  <  y  \/  x (  _I  |`  RR* ) y ) ) )
18 brun 4487 . . . 4  |-  ( x (  <  u.  (  _I  |`  RR* ) ) y  <-> 
( x  <  y  \/  x (  _I  |`  RR* )
y ) )
1917, 18syl6bbr 263 . . 3  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
x  <_  y  <->  x (  <  u.  (  _I  |`  RR* )
) y ) )
2012, 13, 19pm5.21nii 351 . 2  |-  ( x  <_  y  <->  x (  <  u.  (  _I  |`  RR* )
) y )
211, 10, 20eqbrriv 5086 1  |-  <_  =  (  <  u.  (  _I  |`  RR* ) )
Colors of variables: wff setvar class
Syntax hints:    \/ wo 366    /\ wa 367    = wceq 1398    e. wcel 1823    u. cun 3459    C_ wss 3461   class class class wbr 4439    _I cid 4779    X. cxp 4986    |` cres 4990   Rel wrel 4993   -->wf 5566   -1-1-onto->wf1o 5569   RR*cxr 9616    < clt 9617    <_ cle 9618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-pre-lttri 9555  ax-pre-lttrn 9556
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623
This theorem is referenced by: (None)
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