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Theorem suctr 4184
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 108 . . . . 5  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  y  e.  suc  A )
2 vex 2605 . . . . . 6  |-  y  e. 
_V
32elsuc 4169 . . . . 5  |-  ( y  e.  suc  A  <->  ( y  e.  A  \/  y  =  A ) )
41, 3sylib 120 . . . 4  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  ( y  e.  A  \/  y  =  A ) )
5 simpl 107 . . . . . . 7  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  z  e.  y )
6 eleq2 2143 . . . . . . 7  |-  ( y  =  A  ->  (
z  e.  y  <->  z  e.  A ) )
75, 6syl5ibcom 153 . . . . . 6  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  ( y  =  A  ->  z  e.  A ) )
8 elelsuc 4172 . . . . . 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 3890 . . . . . . . . 9  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )
1110expd 254 . . . . . . . 8  |-  ( Tr  A  ->  ( z  e.  y  ->  ( y  e.  A  ->  z  e.  A ) ) )
1211adantrd 273 . . . . . . 7  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  suc  A )  ->  ( y  e.  A  ->  z  e.  A ) ) )
1312, 8syl8 70 . . . . . 6  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  suc  A )  ->  ( y  e.  A  ->  z  e.  suc  A ) ) )
14 jao 705 . . . . . 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 42 . . . 4  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  suc  A )  ->  ( ( y  e.  A  \/  y  =  A )  ->  z  e.  suc  A ) ) )
174, 16mpdi 42 . . 3  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  suc  A )  ->  z  e.  suc  A ) )
1817alrimivv 1797 . 2  |-  ( Tr  A  ->  A. z A. y ( ( z  e.  y  /\  y  e.  suc  A )  -> 
z  e.  suc  A
) )
19 dftr2 3885 . 2  |-  ( Tr 
suc  A  <->  A. z A. y
( ( z  e.  y  /\  y  e. 
suc  A )  -> 
z  e.  suc  A
) )
2018, 19sylibr 132 1  |-  ( Tr  A  ->  Tr  suc  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    \/ wo 662   A.wal 1283    = wceq 1285    e. wcel 1434   Tr wtr 3883   suc csuc 4128
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-sn 3412  df-uni 3610  df-tr 3884  df-suc 4134
This theorem is referenced by:  ordsucim  4252  ordom  4355
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