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Theorem suctrALTcf 32010
Description: The sucessor of a transitive class is transitive. suctrALTcf 32010, using conventional notation, was translated from virtual deduction form, suctrALTcfVD 32011, using a translation program. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
suctrALTcf  |-  ( Tr  A  ->  Tr  suc  A
)

Proof of Theorem suctrALTcf
Dummy variables  z 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sssucid 4907 . . . . . . . 8  |-  A  C_  suc  A
2 id 22 . . . . . . . . 9  |-  ( Tr  A  ->  Tr  A
)
3 id 22 . . . . . . . . . 10  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  ( z  e.  y  /\  y  e. 
suc  A ) )
4 simpl 457 . . . . . . . . . 10  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  z  e.  y )
53, 4syl 16 . . . . . . . . 9  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  z  e.  y )
6 id 22 . . . . . . . . 9  |-  ( y  e.  A  ->  y  e.  A )
7 trel 4503 . . . . . . . . . 10  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )
873impib 1186 . . . . . . . . 9  |-  ( ( Tr  A  /\  z  e.  y  /\  y  e.  A )  ->  z  e.  A )
92, 5, 6, 8syl3an 1261 . . . . . . . 8  |-  ( ( Tr  A  /\  (
z  e.  y  /\  y  e.  suc  A )  /\  y  e.  A
)  ->  z  e.  A )
10 ssel2 3462 . . . . . . . 8  |-  ( ( A  C_  suc  A  /\  z  e.  A )  ->  z  e.  suc  A
)
111, 9, 10eel0321old 31799 . . . . . . 7  |-  ( ( Tr  A  /\  (
z  e.  y  /\  y  e.  suc  A )  /\  y  e.  A
)  ->  z  e.  suc  A )
12113expia 1190 . . . . . 6  |-  ( ( Tr  A  /\  (
z  e.  y  /\  y  e.  suc  A ) )  ->  ( y  e.  A  ->  z  e. 
suc  A ) )
13 id 22 . . . . . . . . 9  |-  ( y  =  A  ->  y  =  A )
14 eleq2 2527 . . . . . . . . . 10  |-  ( y  =  A  ->  (
z  e.  y  <->  z  e.  A ) )
1514biimpac 486 . . . . . . . . 9  |-  ( ( z  e.  y  /\  y  =  A )  ->  z  e.  A )
165, 13, 15syl2an 477 . . . . . . . 8  |-  ( ( ( z  e.  y  /\  y  e.  suc  A )  /\  y  =  A )  ->  z  e.  A )
171, 16, 10eel021old 31774 . . . . . . 7  |-  ( ( ( z  e.  y  /\  y  e.  suc  A )  /\  y  =  A )  ->  z  e.  suc  A )
1817ex 434 . . . . . 6  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  ( y  =  A  ->  z  e.  suc  A ) )
19 simpr 461 . . . . . . . 8  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  y  e.  suc  A )
203, 19syl 16 . . . . . . 7  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  y  e.  suc  A )
21 elsuci 4896 . . . . . . 7  |-  ( y  e.  suc  A  -> 
( y  e.  A  \/  y  =  A
) )
2220, 21syl 16 . . . . . 6  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  ( y  e.  A  \/  y  =  A ) )
23 jao 512 . . . . . . 7  |-  ( ( y  e.  A  -> 
z  e.  suc  A
)  ->  ( (
y  =  A  -> 
z  e.  suc  A
)  ->  ( (
y  e.  A  \/  y  =  A )  ->  z  e.  suc  A
) ) )
24233imp 1182 . . . . . 6  |-  ( ( ( y  e.  A  ->  z  e.  suc  A
)  /\  ( y  =  A  ->  z  e. 
suc  A )  /\  ( y  e.  A  \/  y  =  A
) )  ->  z  e.  suc  A )
2512, 18, 22, 24eel2122old 31801 . . . . 5  |-  ( ( Tr  A  /\  (
z  e.  y  /\  y  e.  suc  A ) )  ->  z  e.  suc  A )
2625ex 434 . . . 4  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  suc  A )  ->  z  e.  suc  A ) )
2726alrimivv 1687 . . 3  |-  ( Tr  A  ->  A. z A. y ( ( z  e.  y  /\  y  e.  suc  A )  -> 
z  e.  suc  A
) )
28 dftr2 4498 . . . 4  |-  ( Tr 
suc  A  <->  A. z A. y
( ( z  e.  y  /\  y  e. 
suc  A )  -> 
z  e.  suc  A
) )
2928biimpri 206 . . 3  |-  ( A. z A. y ( ( z  e.  y  /\  y  e.  suc  A )  ->  z  e.  suc  A )  ->  Tr  suc  A
)
3027, 29syl 16 . 2  |-  ( Tr  A  ->  Tr  suc  A
)
3130iin1 31637 1  |-  ( Tr  A  ->  Tr  suc  A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369   A.wal 1368    = wceq 1370    e. wcel 1758    C_ wss 3439   Tr wtr 4496   suc csuc 4832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3080  df-un 3444  df-in 3446  df-ss 3453  df-sn 3989  df-uni 4203  df-tr 4497  df-suc 4836
This theorem is referenced by: (None)
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