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Theorem predpo 24184
Description: Property of the precessor class for partial orderings. (Contributed by Scott Fenton, 28-Apr-2012.)
Assertion
Ref Expression
predpo  |-  ( ( R  Po  A  /\  X  e.  A )  ->  ( Y  e.  Pred ( R ,  A ,  X )  ->  Pred ( R ,  A ,  Y )  C_  Pred ( R ,  A ,  X ) ) )

Proof of Theorem predpo
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 predel 24183 . 2  |-  ( Y  e.  Pred ( R ,  A ,  X )  ->  Y  e.  A )
2 elpredg 24178 . . . . . . . . . . 11  |-  ( ( X  e.  A  /\  Y  e.  A )  ->  ( Y  e.  Pred ( R ,  A ,  X )  <->  Y R X ) )
32adantll 694 . . . . . . . . . 10  |-  ( ( ( R  Po  A  /\  X  e.  A
)  /\  Y  e.  A )  ->  ( Y  e.  Pred ( R ,  A ,  X
)  <->  Y R X ) )
4 potr 4326 . . . . . . . . . . . . . . . 16  |-  ( ( R  Po  A  /\  ( z  e.  A  /\  Y  e.  A  /\  X  e.  A
) )  ->  (
( z R Y  /\  Y R X )  ->  z R X ) )
543exp2 1169 . . . . . . . . . . . . . . 15  |-  ( R  Po  A  ->  (
z  e.  A  -> 
( Y  e.  A  ->  ( X  e.  A  ->  ( ( z R Y  /\  Y R X )  ->  z R X ) ) ) ) )
65com24 81 . . . . . . . . . . . . . 14  |-  ( R  Po  A  ->  ( X  e.  A  ->  ( Y  e.  A  -> 
( z  e.  A  ->  ( ( z R Y  /\  Y R X )  ->  z R X ) ) ) ) )
76imp31 421 . . . . . . . . . . . . 13  |-  ( ( ( R  Po  A  /\  X  e.  A
)  /\  Y  e.  A )  ->  (
z  e.  A  -> 
( ( z R Y  /\  Y R X )  ->  z R X ) ) )
87com13 74 . . . . . . . . . . . 12  |-  ( ( z R Y  /\  Y R X )  -> 
( z  e.  A  ->  ( ( ( R  Po  A  /\  X  e.  A )  /\  Y  e.  A )  ->  z R X ) ) )
98ex 423 . . . . . . . . . . 11  |-  ( z R Y  ->  ( Y R X  ->  (
z  e.  A  -> 
( ( ( R  Po  A  /\  X  e.  A )  /\  Y  e.  A )  ->  z R X ) ) ) )
109com14 82 . . . . . . . . . 10  |-  ( ( ( R  Po  A  /\  X  e.  A
)  /\  Y  e.  A )  ->  ( Y R X  ->  (
z  e.  A  -> 
( z R Y  ->  z R X ) ) ) )
113, 10sylbid 206 . . . . . . . . 9  |-  ( ( ( R  Po  A  /\  X  e.  A
)  /\  Y  e.  A )  ->  ( Y  e.  Pred ( R ,  A ,  X
)  ->  ( z  e.  A  ->  ( z R Y  ->  z R X ) ) ) )
1211ex 423 . . . . . . . 8  |-  ( ( R  Po  A  /\  X  e.  A )  ->  ( Y  e.  A  ->  ( Y  e.  Pred ( R ,  A ,  X )  ->  (
z  e.  A  -> 
( z R Y  ->  z R X ) ) ) ) )
1312com23 72 . . . . . . 7  |-  ( ( R  Po  A  /\  X  e.  A )  ->  ( Y  e.  Pred ( R ,  A ,  X )  ->  ( Y  e.  A  ->  ( z  e.  A  -> 
( z R Y  ->  z R X ) ) ) ) )
14133imp 1145 . . . . . 6  |-  ( ( ( R  Po  A  /\  X  e.  A
)  /\  Y  e.  Pred ( R ,  A ,  X )  /\  Y  e.  A )  ->  (
z  e.  A  -> 
( z R Y  ->  z R X ) ) )
1514imdistand 673 . . . . 5  |-  ( ( ( R  Po  A  /\  X  e.  A
)  /\  Y  e.  Pred ( R ,  A ,  X )  /\  Y  e.  A )  ->  (
( z  e.  A  /\  z R Y )  ->  ( z  e.  A  /\  z R X ) ) )
16 vex 2791 . . . . . . 7  |-  z  e. 
_V
1716elpred 24177 . . . . . 6  |-  ( Y  e.  A  ->  (
z  e.  Pred ( R ,  A ,  Y )  <->  ( z  e.  A  /\  z R Y ) ) )
18173ad2ant3 978 . . . . 5  |-  ( ( ( R  Po  A  /\  X  e.  A
)  /\  Y  e.  Pred ( R ,  A ,  X )  /\  Y  e.  A )  ->  (
z  e.  Pred ( R ,  A ,  Y )  <->  ( z  e.  A  /\  z R Y ) ) )
1916elpred 24177 . . . . . . 7  |-  ( X  e.  A  ->  (
z  e.  Pred ( R ,  A ,  X )  <->  ( z  e.  A  /\  z R X ) ) )
2019adantl 452 . . . . . 6  |-  ( ( R  Po  A  /\  X  e.  A )  ->  ( z  e.  Pred ( R ,  A ,  X )  <->  ( z  e.  A  /\  z R X ) ) )
21203ad2ant1 976 . . . . 5  |-  ( ( ( R  Po  A  /\  X  e.  A
)  /\  Y  e.  Pred ( R ,  A ,  X )  /\  Y  e.  A )  ->  (
z  e.  Pred ( R ,  A ,  X )  <->  ( z  e.  A  /\  z R X ) ) )
2215, 18, 213imtr4d 259 . . . 4  |-  ( ( ( R  Po  A  /\  X  e.  A
)  /\  Y  e.  Pred ( R ,  A ,  X )  /\  Y  e.  A )  ->  (
z  e.  Pred ( R ,  A ,  Y )  ->  z  e.  Pred ( R ,  A ,  X )
) )
2322ssrdv 3185 . . 3  |-  ( ( ( R  Po  A  /\  X  e.  A
)  /\  Y  e.  Pred ( R ,  A ,  X )  /\  Y  e.  A )  ->  Pred ( R ,  A ,  Y )  C_  Pred ( R ,  A ,  X ) )
24233exp 1150 . 2  |-  ( ( R  Po  A  /\  X  e.  A )  ->  ( Y  e.  Pred ( R ,  A ,  X )  ->  ( Y  e.  A  ->  Pred ( R ,  A ,  Y )  C_  Pred ( R ,  A ,  X ) ) ) )
251, 24mpdi 38 1  |-  ( ( R  Po  A  /\  X  e.  A )  ->  ( Y  e.  Pred ( R ,  A ,  X )  ->  Pred ( R ,  A ,  Y )  C_  Pred ( R ,  A ,  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    e. wcel 1684    C_ wss 3152   class class class wbr 4023    Po wpo 4312   Predcpred 24167
This theorem is referenced by:  predso  24185  trpredpo  24238
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-po 4314  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-pred 24168
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