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Theorem suctrALTcfVD 37360
Description: The following User's Proof is a Virtual Deduction proof (see wvd1 36982) using conjunction-form virtual hypothesis collections. The conjunction-form version of completeusersproof.cmd. It allows the User to avoid superflous virtual hypotheses. This proof was completed automatically by a tools program which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. suctrALTcf 37359 is suctrALTcfVD 37360 without virtual deductions and was derived automatically from suctrALTcfVD 37360. The version of completeusersproof.cmd used is capable of only generating conjunction-form unification theorems, not unification deductions. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
1::  |-  (. Tr  A  ->.  Tr  A ).
2::  |-  (..........  ( z  e.  y  /\  y  e.  suc  A )  ->.  ( z  e.  y  /\  y  e.  suc  A ) ).
3:2:  |-  (..........  ( z  e.  y  /\  y  e.  suc  A )  ->.  z  e.  y ).
4::  |-  (.................................... .......  y  e.  A  ->.  y  e.  A ).
5:1,3,4:  |-  (. (. Tr  A ,. ( z  e.  y  /\  y  e.  suc  A )  ,  y  e.  A ).  ->.  z  e.  A ).
6::  |-  A  C_  suc  A
7:5,6:  |-  (. (. Tr  A ,. ( z  e.  y  /\  y  e.  suc  A )  ,  y  e.  A ).  ->.  z  e.  suc  A ).
8:7:  |-  (. (. Tr  A ,. ( z  e.  y  /\  y  e.  suc  A )  ).  ->.  ( y  e.  A  ->  z  e.  suc  A ) ).
9::  |-  (.................................... ......  y  =  A  ->.  y  =  A ).
10:3,9:  |-  (.........  (. ( z  e.  y  /\  y  e.  suc  A ) ,  y  =  A ).  ->.  z  e.  A ).
11:10,6:  |-  (.........  (. ( z  e.  y  /\  y  e.  suc  A ) ,  y  =  A ).  ->.  z  e.  suc  A ).
12:11:  |-  (...........  ( z  e.  y  /\  y  e.  suc  A )  ->.  ( y  =  A  ->  z  e.  suc  A ) ).
13:2:  |-  (...........  ( z  e.  y  /\  y  e.  suc  A )  ->.  y  e.  suc  A ).
14:13:  |-  (...........  ( z  e.  y  /\  y  e.  suc  A )  ->.  ( y  e.  A  \/  y  =  A ) ).
15:8,12,14:  |-  (. (. Tr  A ,. ( z  e.  y  /\  y  e.  suc  A )  ).  ->.  z  e.  suc  A ).
16:15:  |-  (. Tr  A  ->.  ( ( z  e.  y  /\  y  e.  suc  A )  ->  z  e.  suc  A ) ).
17:16:  |-  (. Tr  A  ->.  A. z A. y ( ( z  e.  y  /\  y  e.  suc  A )  ->  z  e.  suc  A ) ).
18:17:  |-  (. Tr  A  ->.  Tr  suc  A ).
qed:18:  |-  ( Tr  A  ->  Tr  suc  A )
Assertion
Ref Expression
suctrALTcfVD  |-  ( Tr  A  ->  Tr  suc  A
)

Proof of Theorem suctrALTcfVD
Dummy variables  z 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sssucid 5519 . . . . . . . 8  |-  A  C_  suc  A
2 idn1 36987 . . . . . . . . 9  |-  (. Tr  A 
->.  Tr  A ).
3 idn1 36987 . . . . . . . . . 10  |-  (. (
z  e.  y  /\  y  e.  suc  A )  ->.  ( z  e.  y  /\  y  e.  suc  A ) ).
4 simpl 463 . . . . . . . . . 10  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  z  e.  y )
53, 4el1 37050 . . . . . . . . 9  |-  (. (
z  e.  y  /\  y  e.  suc  A )  ->.  z  e.  y ).
6 idn1 36987 . . . . . . . . 9  |-  (. y  e.  A  ->.  y  e.  A ).
7 trel 4518 . . . . . . . . . 10  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )
873impib 1213 . . . . . . . . 9  |-  ( ( Tr  A  /\  z  e.  y  /\  y  e.  A )  ->  z  e.  A )
92, 5, 6, 8el123 37191 . . . . . . . 8  |-  (. (. Tr  A ,. ( z  e.  y  /\  y  e.  suc  A ) ,. y  e.  A ).  ->.  z  e.  A ).
10 ssel2 3439 . . . . . . . 8  |-  ( ( A  C_  suc  A  /\  z  e.  A )  ->  z  e.  suc  A
)
111, 9, 10el0321old 37142 . . . . . . 7  |-  (. (. Tr  A ,. ( z  e.  y  /\  y  e.  suc  A ) ,. y  e.  A ).  ->.  z  e.  suc  A ).
1211int3 37034 . . . . . 6  |-  (. (. Tr  A ,. ( z  e.  y  /\  y  e.  suc  A ) ).  ->.  ( y  e.  A  ->  z  e.  suc  A
) ).
13 idn1 36987 . . . . . . . . 9  |-  (. y  =  A  ->.  y  =  A ).
14 eleq2 2529 . . . . . . . . . 10  |-  ( y  =  A  ->  (
z  e.  y  <->  z  e.  A ) )
1514biimpac 493 . . . . . . . . 9  |-  ( ( z  e.  y  /\  y  =  A )  ->  z  e.  A )
165, 13, 15el12 37153 . . . . . . . 8  |-  (. (. ( z  e.  y  /\  y  e.  suc  A ) ,. y  =  A ).  ->.  z  e.  A ).
171, 16, 10el021old 37123 . . . . . . 7  |-  (. (. ( z  e.  y  /\  y  e.  suc  A ) ,. y  =  A ).  ->.  z  e.  suc  A ).
1817int2 37028 . . . . . 6  |-  (. (
z  e.  y  /\  y  e.  suc  A )  ->.  ( y  =  A  ->  z  e.  suc  A ) ).
19 simpr 467 . . . . . . . 8  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  y  e.  suc  A )
203, 19el1 37050 . . . . . . 7  |-  (. (
z  e.  y  /\  y  e.  suc  A )  ->.  y  e.  suc  A ).
21 elsuci 5508 . . . . . . 7  |-  ( y  e.  suc  A  -> 
( y  e.  A  \/  y  =  A
) )
2220, 21el1 37050 . . . . . 6  |-  (. (
z  e.  y  /\  y  e.  suc  A )  ->.  ( y  e.  A  \/  y  =  A
) ).
23 jao 519 . . . . . . 7  |-  ( ( y  e.  A  -> 
z  e.  suc  A
)  ->  ( (
y  =  A  -> 
z  e.  suc  A
)  ->  ( (
y  e.  A  \/  y  =  A )  ->  z  e.  suc  A
) ) )
24233imp 1208 . . . . . 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, 24el2122old 37144 . . . . 5  |-  (. (. Tr  A ,. ( z  e.  y  /\  y  e.  suc  A ) ).  ->.  z  e.  suc  A ).
2625int2 37028 . . . 4  |-  (. Tr  A 
->.  ( ( z  e.  y  /\  y  e. 
suc  A )  -> 
z  e.  suc  A
) ).
2726gen12 37040 . . 3  |-  (. Tr  A 
->.  A. z A. y
( ( z  e.  y  /\  y  e. 
suc  A )  -> 
z  e.  suc  A
) ).
28 dftr2 4513 . . . 4  |-  ( Tr 
suc  A  <->  A. z A. y
( ( z  e.  y  /\  y  e. 
suc  A )  -> 
z  e.  suc  A
) )
2928biimpri 211 . . 3  |-  ( A. z A. y ( ( z  e.  y  /\  y  e.  suc  A )  ->  z  e.  suc  A )  ->  Tr  suc  A
)
3027, 29el1 37050 . 2  |-  (. Tr  A 
->.  Tr  suc  A ).
3130in1 36984 1  |-  ( Tr  A  ->  Tr  suc  A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 374    /\ wa 375   A.wal 1453    = wceq 1455    e. wcel 1898    C_ wss 3416   Tr wtr 4511   suc csuc 5444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-v 3059  df-un 3421  df-in 3423  df-ss 3430  df-sn 3981  df-uni 4213  df-tr 4512  df-suc 5448  df-vd1 36983  df-vhc2 36994  df-vhc3 37002
This theorem is referenced by: (None)
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