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Theorem suctrALT2 37233
Description: Virtual deduction proof of suctr 5506. The sucessor of a transitive class is transitive. This proof was generated automatically from the virtual deduction proof suctrALT2VD 37232 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 5500 . . . . 5  |-  A  C_  suc  A
2 trel 4504 . . . . . . 7  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )
32expd 438 . . . . . 6  |-  ( Tr  A  ->  ( z  e.  y  ->  ( y  e.  A  ->  z  e.  A ) ) )
43adantrd 470 . . . . 5  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  suc  A )  ->  ( y  e.  A  ->  z  e.  A ) ) )
5 ssel 3426 . . . . 5  |-  ( A 
C_  suc  A  ->  ( z  e.  A  -> 
z  e.  suc  A
) )
61, 4, 5ee03 37128 . . . 4  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  suc  A )  ->  ( y  e.  A  ->  z  e.  suc  A ) ) )
7 simpl 459 . . . . . . 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 2518 . . . . . . 7  |-  ( y  =  A  ->  (
z  e.  y  <->  z  e.  A ) )
109biimpcd 228 . . . . . 6  |-  ( z  e.  y  ->  (
y  =  A  -> 
z  e.  A ) )
118, 10syl6 34 . . . . 5  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  suc  A )  ->  ( y  =  A  ->  z  e.  A ) ) )
121, 11, 5ee03 37128 . . . 4  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  suc  A )  ->  ( y  =  A  ->  z  e.  suc  A ) ) )
13 simpr 463 . . . . . 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 5489 . . . . 5  |-  ( y  e.  suc  A  -> 
( y  e.  A  \/  y  =  A
) )
1614, 15syl6 34 . . . 4  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  suc  A )  ->  ( y  e.  A  \/  y  =  A ) ) )
17 jao 515 . . . 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 36858 . . 3  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  suc  A )  ->  z  e.  suc  A ) )
1918alrimivv 1774 . 2  |-  ( Tr  A  ->  A. z A. y ( ( z  e.  y  /\  y  e.  suc  A )  -> 
z  e.  suc  A
) )
20 dftr2 4499 . 2  |-  ( Tr 
suc  A  <->  A. z A. y
( ( z  e.  y  /\  y  e. 
suc  A )  -> 
z  e.  suc  A
) )
2119, 20sylibr 216 1  |-  ( Tr  A  ->  Tr  suc  A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 370    /\ wa 371   A.wal 1442    = wceq 1444    e. wcel 1887    C_ wss 3404   Tr wtr 4497   suc csuc 5425
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-v 3047  df-un 3409  df-in 3411  df-ss 3418  df-sn 3969  df-uni 4199  df-tr 4498  df-suc 5429
This theorem is referenced by: (None)
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