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Theorem predpo 25406
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 25405 . 2  |-  ( Y  e.  Pred ( R ,  A ,  X )  ->  Y  e.  A )
2 elpredg 25400 . . . . . . . . . . 11  |-  ( ( X  e.  A  /\  Y  e.  A )  ->  ( Y  e.  Pred ( R ,  A ,  X )  <->  Y R X ) )
32adantll 695 . . . . . . . . . 10  |-  ( ( ( R  Po  A  /\  X  e.  A
)  /\  Y  e.  A )  ->  ( Y  e.  Pred ( R ,  A ,  X
)  <->  Y R X ) )
4 potr 4483 . . . . . . . . . . . . . . . 16  |-  ( ( R  Po  A  /\  ( z  e.  A  /\  Y  e.  A  /\  X  e.  A
) )  ->  (
( z R Y  /\  Y R X )  ->  z R X ) )
543exp2 1171 . . . . . . . . . . . . . . 15  |-  ( R  Po  A  ->  (
z  e.  A  -> 
( Y  e.  A  ->  ( X  e.  A  ->  ( ( z R Y  /\  Y R X )  ->  z R X ) ) ) ) )
65com24 83 . . . . . . . . . . . . . 14  |-  ( R  Po  A  ->  ( X  e.  A  ->  ( Y  e.  A  -> 
( z  e.  A  ->  ( ( z R Y  /\  Y R X )  ->  z R X ) ) ) ) )
76imp31 422 . . . . . . . . . . . . 13  |-  ( ( ( R  Po  A  /\  X  e.  A
)  /\  Y  e.  A )  ->  (
z  e.  A  -> 
( ( z R Y  /\  Y R X )  ->  z R X ) ) )
87com13 76 . . . . . . . . . . . 12  |-  ( ( z R Y  /\  Y R X )  -> 
( z  e.  A  ->  ( ( ( R  Po  A  /\  X  e.  A )  /\  Y  e.  A )  ->  z R X ) ) )
98ex 424 . . . . . . . . . . 11  |-  ( z R Y  ->  ( Y R X  ->  (
z  e.  A  -> 
( ( ( R  Po  A  /\  X  e.  A )  /\  Y  e.  A )  ->  z R X ) ) ) )
109com14 84 . . . . . . . . . 10  |-  ( ( ( R  Po  A  /\  X  e.  A
)  /\  Y  e.  A )  ->  ( Y R X  ->  (
z  e.  A  -> 
( z R Y  ->  z R X ) ) ) )
113, 10sylbid 207 . . . . . . . . 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 424 . . . . . . . 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 74 . . . . . . 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 1147 . . . . . 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 674 . . . . 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 2927 . . . . . . 7  |-  z  e. 
_V
1716elpred 25399 . . . . . 6  |-  ( Y  e.  A  ->  (
z  e.  Pred ( R ,  A ,  Y )  <->  ( z  e.  A  /\  z R Y ) ) )
18173ad2ant3 980 . . . . 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 25399 . . . . . . 7  |-  ( X  e.  A  ->  (
z  e.  Pred ( R ,  A ,  X )  <->  ( z  e.  A  /\  z R X ) ) )
2019adantl 453 . . . . . 6  |-  ( ( R  Po  A  /\  X  e.  A )  ->  ( z  e.  Pred ( R ,  A ,  X )  <->  ( z  e.  A  /\  z R X ) ) )
21203ad2ant1 978 . . . . 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 260 . . . 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 3322 . . 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 1152 . 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 40 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 177    /\ wa 359    /\ w3a 936    e. wcel 1721    C_ wss 3288   class class class wbr 4180    Po wpo 4469   Predcpred 25389
This theorem is referenced by:  predso  25407  trpredpo  25460
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-br 4181  df-opab 4235  df-po 4471  df-xp 4851  df-cnv 4853  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 25390
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