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Theorem suctrALT2 28850
Description: Virtual deduction proof of suctr 4657. The sucessor of a transitive class is transitive. This proof was generated automatically from the virtual deduction proof suctrALT2VD 28849 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 4650 . . . . 5  |-  A  C_  suc  A
2 trel 4301 . . . . . . 7  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )
32exp3a 426 . . . . . 6  |-  ( Tr  A  ->  ( z  e.  y  ->  ( y  e.  A  ->  z  e.  A ) ) )
43adantrd 455 . . . . 5  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  suc  A )  ->  ( y  e.  A  ->  z  e.  A ) ) )
5 ssel 3334 . . . . 5  |-  ( A 
C_  suc  A  ->  ( z  e.  A  -> 
z  e.  suc  A
) )
61, 4, 5ee03 28754 . . . 4  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  suc  A )  ->  ( y  e.  A  ->  z  e.  suc  A ) ) )
7 simpl 444 . . . . . . 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 2496 . . . . . . 7  |-  ( y  =  A  ->  (
z  e.  y  <->  z  e.  A ) )
109biimpcd 216 . . . . . 6  |-  ( z  e.  y  ->  (
y  =  A  -> 
z  e.  A ) )
118, 10syl6 31 . . . . 5  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  suc  A )  ->  ( y  =  A  ->  z  e.  A ) ) )
121, 11, 5ee03 28754 . . . 4  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  suc  A )  ->  ( y  =  A  ->  z  e.  suc  A ) ) )
13 simpr 448 . . . . . 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 4639 . . . . 5  |-  ( y  e.  suc  A  -> 
( y  e.  A  \/  y  =  A
) )
1614, 15syl6 31 . . . 4  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  suc  A )  ->  ( y  e.  A  \/  y  =  A ) ) )
17 jao 499 . . . 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 28485 . . 3  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  suc  A )  ->  z  e.  suc  A ) )
1918alrimivv 1642 . 2  |-  ( Tr  A  ->  A. z A. y ( ( z  e.  y  /\  y  e.  suc  A )  -> 
z  e.  suc  A
) )
20 dftr2 4296 . 2  |-  ( Tr 
suc  A  <->  A. z A. y
( ( z  e.  y  /\  y  e. 
suc  A )  -> 
z  e.  suc  A
) )
2119, 20sylibr 204 1  |-  ( Tr  A  ->  Tr  suc  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359   A.wal 1549    = wceq 1652    e. wcel 1725    C_ wss 3312   Tr wtr 4294   suc csuc 4575
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-un 3317  df-in 3319  df-ss 3326  df-sn 3812  df-uni 4008  df-tr 4295  df-suc 4579
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