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Theorem wlogle 9560
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 9559 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 1725    C_ wss 3320   class class class wbr 4212   RRcr 8989    <_ cle 9121
This theorem is referenced by:  vdwlem12  13360  iundisj2  19443  volcn  19498  dvlip  19877  ftc1a  19921  iundisj2f  24030  iundisj2fi  24153  erdszelem9  24885  ftc1anc  26288
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047  ax-pre-lttri 9064
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126
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