MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wlogle Unicode version

Theorem wlogle 9494
Description: If the predicate  ch (
x ,  y ) is symmetric under interchange of  x ,  y, then "without loss of generality" we can assume that  x  <_  y. (Contributed by Mario Carneiro, 18-Aug-2014.) (Revised by Mario Carneiro, 11-Sep-2014.)
Hypotheses
Ref Expression
wlogle.1  |-  ( ( z  =  x  /\  w  =  y )  ->  ( ps  <->  ch )
)
wlogle.2  |-  ( ( z  =  y  /\  w  =  x )  ->  ( ps  <->  th )
)
wlogle.3  |-  ( ph  ->  S  C_  RR )
wlogle.4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( ch  <->  th )
)
wlogle.5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  x  <_  y ) )  ->  ch )
Assertion
Ref Expression
wlogle  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ch )
Distinct variable groups:    x, w, y, z, ph    w, S, x, y, z    ps, x, y    ch, w, z
Allowed substitution hints:    ps( z, w)    ch( x, y)    th( x, y, z, w)

Proof of Theorem wlogle
StepHypRef Expression
1 wlogle.1 . 2  |-  ( ( z  =  x  /\  w  =  y )  ->  ( ps  <->  ch )
)
2 wlogle.2 . 2  |-  ( ( z  =  y  /\  w  =  x )  ->  ( ps  <->  th )
)
3 wlogle.3 . 2  |-  ( ph  ->  S  C_  RR )
4 wlogle.5 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  x  <_  y ) )  ->  ch )
5 wlogle.4 . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( ch  <->  th )
)
653adantr3 1118 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  x  <_  y ) )  -> 
( ch  <->  th )
)
74, 6mpbid 202 . 2  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  x  <_  y ) )  ->  th )
81, 2, 3, 7, 4wloglei 9493 1  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    e. wcel 1717    C_ wss 3265   class class class wbr 4155   RRcr 8924    <_ cle 9056
This theorem is referenced by:  vdwlem12  13289  iundisj2  19312  volcn  19367  dvlip  19746  ftc1a  19790  iundisj2f  23875  iundisj2fi  23993  erdszelem9  24666
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-resscn 8982  ax-pre-lttri 8999
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061
  Copyright terms: Public domain W3C validator