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Theorem suctr 4338
Description: The successor of a transitive class is transitive. (Contributed by Alan Sare, 11-Apr-2009.)
Assertion
Ref Expression
suctr  |-  ( Tr  A  ->  Tr  suc  A
)

Proof of Theorem suctr
Dummy variables  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 109 . . . . 5  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  y  e.  suc  A )
2 vex 2684 . . . . . 6  |-  y  e. 
_V
32elsuc 4323 . . . . 5  |-  ( y  e.  suc  A  <->  ( y  e.  A  \/  y  =  A ) )
41, 3sylib 121 . . . 4  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  ( y  e.  A  \/  y  =  A ) )
5 simpl 108 . . . . . . 7  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  z  e.  y )
6 eleq2 2201 . . . . . . 7  |-  ( y  =  A  ->  (
z  e.  y  <->  z  e.  A ) )
75, 6syl5ibcom 154 . . . . . 6  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  ( y  =  A  ->  z  e.  A ) )
8 elelsuc 4326 . . . . . 6  |-  ( z  e.  A  ->  z  e.  suc  A )
97, 8syl6 33 . . . . 5  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  ( y  =  A  ->  z  e.  suc  A ) )
10 trel 4028 . . . . . . . . 9  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )
1110expd 256 . . . . . . . 8  |-  ( Tr  A  ->  ( z  e.  y  ->  ( y  e.  A  ->  z  e.  A ) ) )
1211adantrd 277 . . . . . . 7  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  suc  A )  ->  ( y  e.  A  ->  z  e.  A ) ) )
1312, 8syl8 71 . . . . . 6  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  suc  A )  ->  ( y  e.  A  ->  z  e.  suc  A ) ) )
14 jao 744 . . . . . 6  |-  ( ( y  e.  A  -> 
z  e.  suc  A
)  ->  ( (
y  =  A  -> 
z  e.  suc  A
)  ->  ( (
y  e.  A  \/  y  =  A )  ->  z  e.  suc  A
) ) )
1513, 14syl6 33 . . . . 5  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  suc  A )  ->  ( ( y  =  A  ->  z  e.  suc  A )  -> 
( ( y  e.  A  \/  y  =  A )  ->  z  e.  suc  A ) ) ) )
169, 15mpdi 43 . . . 4  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  suc  A )  ->  ( ( y  e.  A  \/  y  =  A )  ->  z  e.  suc  A ) ) )
174, 16mpdi 43 . . 3  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  suc  A )  ->  z  e.  suc  A ) )
1817alrimivv 1847 . 2  |-  ( Tr  A  ->  A. z A. y ( ( z  e.  y  /\  y  e.  suc  A )  -> 
z  e.  suc  A
) )
19 dftr2 4023 . 2  |-  ( Tr 
suc  A  <->  A. z A. y
( ( z  e.  y  /\  y  e. 
suc  A )  -> 
z  e.  suc  A
) )
2018, 19sylibr 133 1  |-  ( Tr  A  ->  Tr  suc  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 697   A.wal 1329    = wceq 1331    e. wcel 1480   Tr wtr 4021   suc csuc 4282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-sn 3528  df-uni 3732  df-tr 4022  df-suc 4288
This theorem is referenced by:  ordsucim  4411  ordom  4515
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