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Theorem predpo 25010
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 25009 . 2  |-  ( Y  e.  Pred ( R ,  A ,  X )  ->  Y  e.  A )
2 elpredg 25004 . . . . . . . . . . 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 4429 . . . . . . . . . . . . . . . 16  |-  ( ( R  Po  A  /\  ( z  e.  A  /\  Y  e.  A  /\  X  e.  A
) )  ->  (
( z R Y  /\  Y R X )  ->  z R X ) )
543exp2 1170 . . . . . . . . . . . . . . 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 1146 . . . . . 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 2876 . . . . . . 7  |-  z  e. 
_V
1716elpred 25003 . . . . . 6  |-  ( Y  e.  A  ->  (
z  e.  Pred ( R ,  A ,  Y )  <->  ( z  e.  A  /\  z R Y ) ) )
18173ad2ant3 979 . . . . 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 25003 . . . . . . 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 977 . . . . 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 3271 . . 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 1151 . 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 935    e. wcel 1715    C_ wss 3238   class class class wbr 4125    Po wpo 4415   Predcpred 24993
This theorem is referenced by:  predso  25011  trpredpo  25064
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-br 4126  df-opab 4180  df-po 4417  df-xp 4798  df-cnv 4800  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-pred 24994
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