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Theorem suctr 5505
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 463 . . . . 5  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  y  e.  suc  A )
2 vex 3047 . . . . . 6  |-  y  e. 
_V
32elsuc 5491 . . . . 5  |-  ( y  e.  suc  A  <->  ( y  e.  A  \/  y  =  A ) )
41, 3sylib 200 . . . 4  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  ( y  e.  A  \/  y  =  A ) )
5 simpl 459 . . . . . . 7  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  z  e.  y )
6 eleq2 2517 . . . . . . 7  |-  ( y  =  A  ->  (
z  e.  y  <->  z  e.  A ) )
75, 6syl5ibcom 224 . . . . . 6  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  ( y  =  A  ->  z  e.  A ) )
8 elelsuc 5494 . . . . . 6  |-  ( z  e.  A  ->  z  e.  suc  A )
97, 8syl6 34 . . . . 5  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  ( y  =  A  ->  z  e.  suc  A ) )
10 trel 4503 . . . . . . . . 9  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )
1110expd 438 . . . . . . . 8  |-  ( Tr  A  ->  ( z  e.  y  ->  ( y  e.  A  ->  z  e.  A ) ) )
1211adantrd 470 . . . . . . 7  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  suc  A )  ->  ( y  e.  A  ->  z  e.  A ) ) )
1312, 8syl8 72 . . . . . 6  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  suc  A )  ->  ( y  e.  A  ->  z  e.  suc  A ) ) )
14 jao 515 . . . . . 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 34 . . . . 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 1773 . 2  |-  ( Tr  A  ->  A. z A. y ( ( z  e.  y  /\  y  e.  suc  A )  -> 
z  e.  suc  A
) )
19 dftr2 4498 . 2  |-  ( Tr 
suc  A  <->  A. z A. y
( ( z  e.  y  /\  y  e. 
suc  A )  -> 
z  e.  suc  A
) )
2018, 19sylibr 216 1  |-  ( Tr  A  ->  Tr  suc  A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 370    /\ wa 371   A.wal 1441    = wceq 1443    e. wcel 1886   Tr wtr 4496   suc csuc 5424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-v 3046  df-un 3408  df-in 3410  df-ss 3417  df-sn 3968  df-uni 4198  df-tr 4497  df-suc 5428
This theorem is referenced by:  dfon2lem3  30424  dfon2lem7  30428  dford3lem2  35876
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