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Theorem predpo 28857
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 28856 . 2  |-  ( Y  e.  Pred ( R ,  A ,  X )  ->  Y  e.  A )
2 elpredg 28851 . . . . . . . . . . 11  |-  ( ( X  e.  A  /\  Y  e.  A )  ->  ( Y  e.  Pred ( R ,  A ,  X )  <->  Y R X ) )
32adantll 713 . . . . . . . . . 10  |-  ( ( ( R  Po  A  /\  X  e.  A
)  /\  Y  e.  A )  ->  ( Y  e.  Pred ( R ,  A ,  X
)  <->  Y R X ) )
4 potr 4812 . . . . . . . . . . . . . . . 16  |-  ( ( R  Po  A  /\  ( z  e.  A  /\  Y  e.  A  /\  X  e.  A
) )  ->  (
( z R Y  /\  Y R X )  ->  z R X ) )
543exp2 1214 . . . . . . . . . . . . . . 15  |-  ( R  Po  A  ->  (
z  e.  A  -> 
( Y  e.  A  ->  ( X  e.  A  ->  ( ( z R Y  /\  Y R X )  ->  z R X ) ) ) ) )
65com24 87 . . . . . . . . . . . . . 14  |-  ( R  Po  A  ->  ( X  e.  A  ->  ( Y  e.  A  -> 
( z  e.  A  ->  ( ( z R Y  /\  Y R X )  ->  z R X ) ) ) ) )
76imp31 432 . . . . . . . . . . . . 13  |-  ( ( ( R  Po  A  /\  X  e.  A
)  /\  Y  e.  A )  ->  (
z  e.  A  -> 
( ( z R Y  /\  Y R X )  ->  z R X ) ) )
87com13 80 . . . . . . . . . . . 12  |-  ( ( z R Y  /\  Y R X )  -> 
( z  e.  A  ->  ( ( ( R  Po  A  /\  X  e.  A )  /\  Y  e.  A )  ->  z R X ) ) )
98ex 434 . . . . . . . . . . 11  |-  ( z R Y  ->  ( Y R X  ->  (
z  e.  A  -> 
( ( ( R  Po  A  /\  X  e.  A )  /\  Y  e.  A )  ->  z R X ) ) ) )
109com14 88 . . . . . . . . . 10  |-  ( ( ( R  Po  A  /\  X  e.  A
)  /\  Y  e.  A )  ->  ( Y R X  ->  (
z  e.  A  -> 
( z R Y  ->  z R X ) ) ) )
113, 10sylbid 215 . . . . . . . . 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 434 . . . . . . . 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 78 . . . . . . 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 1190 . . . . . 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 692 . . . . 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 3116 . . . . . . 7  |-  z  e. 
_V
1716elpred 28850 . . . . . 6  |-  ( Y  e.  A  ->  (
z  e.  Pred ( R ,  A ,  Y )  <->  ( z  e.  A  /\  z R Y ) ) )
18173ad2ant3 1019 . . . . 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 28850 . . . . . . 7  |-  ( X  e.  A  ->  (
z  e.  Pred ( R ,  A ,  X )  <->  ( z  e.  A  /\  z R X ) ) )
2019adantl 466 . . . . . 6  |-  ( ( R  Po  A  /\  X  e.  A )  ->  ( z  e.  Pred ( R ,  A ,  X )  <->  ( z  e.  A  /\  z R X ) ) )
21203ad2ant1 1017 . . . . 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 268 . . . 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 3510 . . 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 1195 . 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 42 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    e. wcel 1767    C_ wss 3476   class class class wbr 4447    Po wpo 4798   Predcpred 28836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-po 4800  df-xp 5005  df-cnv 5007  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-pred 28837
This theorem is referenced by:  predso  28858  trpredpo  28911
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