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Theorem predpo 25464
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 25463 . 2  |-  ( Y  e.  Pred ( R ,  A ,  X )  ->  Y  e.  A )
2 elpredg 25458 . . . . . . . . . . 11  |-  ( ( X  e.  A  /\  Y  e.  A )  ->  ( Y  e.  Pred ( R ,  A ,  X )  <->  Y R X ) )
32adantll 696 . . . . . . . . . 10  |-  ( ( ( R  Po  A  /\  X  e.  A
)  /\  Y  e.  A )  ->  ( Y  e.  Pred ( R ,  A ,  X
)  <->  Y R X ) )
4 potr 4518 . . . . . . . . . . . . . . . 16  |-  ( ( R  Po  A  /\  ( z  e.  A  /\  Y  e.  A  /\  X  e.  A
) )  ->  (
( z R Y  /\  Y R X )  ->  z R X ) )
543exp2 1172 . . . . . . . . . . . . . . 15  |-  ( R  Po  A  ->  (
z  e.  A  -> 
( Y  e.  A  ->  ( X  e.  A  ->  ( ( z R Y  /\  Y R X )  ->  z R X ) ) ) ) )
65com24 84 . . . . . . . . . . . . . 14  |-  ( R  Po  A  ->  ( X  e.  A  ->  ( Y  e.  A  -> 
( z  e.  A  ->  ( ( z R Y  /\  Y R X )  ->  z R X ) ) ) ) )
76imp31 423 . . . . . . . . . . . . 13  |-  ( ( ( R  Po  A  /\  X  e.  A
)  /\  Y  e.  A )  ->  (
z  e.  A  -> 
( ( z R Y  /\  Y R X )  ->  z R X ) ) )
87com13 77 . . . . . . . . . . . 12  |-  ( ( z R Y  /\  Y R X )  -> 
( z  e.  A  ->  ( ( ( R  Po  A  /\  X  e.  A )  /\  Y  e.  A )  ->  z R X ) ) )
98ex 425 . . . . . . . . . . 11  |-  ( z R Y  ->  ( Y R X  ->  (
z  e.  A  -> 
( ( ( R  Po  A  /\  X  e.  A )  /\  Y  e.  A )  ->  z R X ) ) ) )
109com14 85 . . . . . . . . . 10  |-  ( ( ( R  Po  A  /\  X  e.  A
)  /\  Y  e.  A )  ->  ( Y R X  ->  (
z  e.  A  -> 
( z R Y  ->  z R X ) ) ) )
113, 10sylbid 208 . . . . . . . . 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 425 . . . . . . . 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 75 . . . . . . 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 1148 . . . . . 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 675 . . . . 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 2961 . . . . . . 7  |-  z  e. 
_V
1716elpred 25457 . . . . . 6  |-  ( Y  e.  A  ->  (
z  e.  Pred ( R ,  A ,  Y )  <->  ( z  e.  A  /\  z R Y ) ) )
18173ad2ant3 981 . . . . 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 25457 . . . . . . 7  |-  ( X  e.  A  ->  (
z  e.  Pred ( R ,  A ,  X )  <->  ( z  e.  A  /\  z R X ) ) )
2019adantl 454 . . . . . 6  |-  ( ( R  Po  A  /\  X  e.  A )  ->  ( z  e.  Pred ( R ,  A ,  X )  <->  ( z  e.  A  /\  z R X ) ) )
21203ad2ant1 979 . . . . 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 261 . . . 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 3356 . . 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 1153 . 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 41 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 178    /\ wa 360    /\ w3a 937    e. wcel 1726    C_ wss 3322   class class class wbr 4215    Po wpo 4504   Predcpred 25443
This theorem is referenced by:  predso  25465  trpredpo  25518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-po 4506  df-xp 4887  df-cnv 4889  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-pred 25444
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