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Theorem suctrALT2 33780
Description: Virtual deduction proof of suctr 4970. The sucessor of a transitive class is transitive. This proof was generated automatically from the virtual deduction proof suctrALT2VD 33779 using the tools command file translatewithout_overwritingminimize_excludingduplicates.cmd . (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
suctrALT2  |-  ( Tr  A  ->  Tr  suc  A
)

Proof of Theorem suctrALT2
Dummy variables  z 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sssucid 4964 . . . . 5  |-  A  C_  suc  A
2 trel 4557 . . . . . . 7  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )
32expd 436 . . . . . 6  |-  ( Tr  A  ->  ( z  e.  y  ->  ( y  e.  A  ->  z  e.  A ) ) )
43adantrd 468 . . . . 5  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  suc  A )  ->  ( y  e.  A  ->  z  e.  A ) ) )
5 ssel 3493 . . . . 5  |-  ( A 
C_  suc  A  ->  ( z  e.  A  -> 
z  e.  suc  A
) )
61, 4, 5ee03 33681 . . . 4  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  suc  A )  ->  ( y  e.  A  ->  z  e.  suc  A ) ) )
7 simpl 457 . . . . . . 7  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  z  e.  y )
87a1i 11 . . . . . 6  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  suc  A )  ->  z  e.  y ) )
9 eleq2 2530 . . . . . . 7  |-  ( y  =  A  ->  (
z  e.  y  <->  z  e.  A ) )
109biimpcd 224 . . . . . 6  |-  ( z  e.  y  ->  (
y  =  A  -> 
z  e.  A ) )
118, 10syl6 33 . . . . 5  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  suc  A )  ->  ( y  =  A  ->  z  e.  A ) ) )
121, 11, 5ee03 33681 . . . 4  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  suc  A )  ->  ( y  =  A  ->  z  e.  suc  A ) ) )
13 simpr 461 . . . . . 6  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  y  e.  suc  A )
1413a1i 11 . . . . 5  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  suc  A )  ->  y  e.  suc  A ) )
15 elsuci 4953 . . . . 5  |-  ( y  e.  suc  A  -> 
( y  e.  A  \/  y  =  A
) )
1614, 15syl6 33 . . . 4  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  suc  A )  ->  ( y  e.  A  \/  y  =  A ) ) )
17 jao 512 . . . 4  |-  ( ( y  e.  A  -> 
z  e.  suc  A
)  ->  ( (
y  =  A  -> 
z  e.  suc  A
)  ->  ( (
y  e.  A  \/  y  =  A )  ->  z  e.  suc  A
) ) )
186, 12, 16, 17ee222 33414 . . 3  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  suc  A )  ->  z  e.  suc  A ) )
1918alrimivv 1721 . 2  |-  ( Tr  A  ->  A. z A. y ( ( z  e.  y  /\  y  e.  suc  A )  -> 
z  e.  suc  A
) )
20 dftr2 4552 . 2  |-  ( Tr 
suc  A  <->  A. z A. y
( ( z  e.  y  /\  y  e. 
suc  A )  -> 
z  e.  suc  A
) )
2119, 20sylibr 212 1  |-  ( Tr  A  ->  Tr  suc  A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369   A.wal 1393    = wceq 1395    e. wcel 1819    C_ wss 3471   Tr wtr 4550   suc csuc 4889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-un 3476  df-in 3478  df-ss 3485  df-sn 4033  df-uni 4252  df-tr 4551  df-suc 4893
This theorem is referenced by: (None)
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