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Theorem wlogle 9260
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 1121 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  x  <_  y ) )  -> 
( ch  <->  th )
)
74, 6mpbid 203 . 2  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  x  <_  y ) )  ->  th )
81, 2, 3, 7, 4wloglei 9259 1  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    e. wcel 1621    C_ wss 3113   class class class wbr 3983   RRcr 8690    <_ cle 8822
This theorem is referenced by:  vdwlem12  12987  iundisj2  18854  volcn  18909  dvlip  19288  ftc1a  19332  erdszelem9  23088
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-resscn 8748  ax-pre-lttri 8765
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-er 6614  df-en 6818  df-dom 6819  df-sdom 6820  df-pnf 8823  df-mnf 8824  df-xr 8825  df-ltxr 8826  df-le 8827
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