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Theorem sspwtrALT2 28913
Description: Short predicate calculus proof of the right-to-left implication of dftr4 4134. A class which is a subclass of its power class is transitive. This proof was constructed by applying Metamath's minimize command to the proof of sspwtrALT 28912, which is the virtual deduction proof sspwtr 28911 without virtual deductions. (Contributed by Alan Sare, 3-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sspwtrALT2  |-  ( A 
C_  ~P A  ->  Tr  A )

Proof of Theorem sspwtrALT2
Dummy variables  z 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 3187 . . . . . 6  |-  ( A 
C_  ~P A  ->  (
y  e.  A  -> 
y  e.  ~P A
) )
21adantld 453 . . . . 5  |-  ( A 
C_  ~P A  ->  (
( z  e.  y  /\  y  e.  A
)  ->  y  e.  ~P A ) )
3 elpwi 3646 . . . . 5  |-  ( y  e.  ~P A  -> 
y  C_  A )
42, 3syl6 29 . . . 4  |-  ( A 
C_  ~P A  ->  (
( z  e.  y  /\  y  e.  A
)  ->  y  C_  A ) )
5 simpl 443 . . . . 5  |-  ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  y )
65a1i 10 . . . 4  |-  ( A 
C_  ~P A  ->  (
( z  e.  y  /\  y  e.  A
)  ->  z  e.  y ) )
7 ssel 3187 . . . 4  |-  ( y 
C_  A  ->  (
z  e.  y  -> 
z  e.  A ) )
84, 6, 7ee22 1352 . . 3  |-  ( A 
C_  ~P A  ->  (
( z  e.  y  /\  y  e.  A
)  ->  z  e.  A ) )
98alrimivv 1622 . 2  |-  ( A 
C_  ~P A  ->  A. z A. y ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )
10 dftr2 4131 . 2  |-  ( Tr  A  <->  A. z A. y
( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )
119, 10sylibr 203 1  |-  ( A 
C_  ~P A  ->  Tr  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1530    e. wcel 1696    C_ wss 3165   ~Pcpw 3638   Tr wtr 4129
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-in 3172  df-ss 3179  df-pw 3640  df-uni 3844  df-tr 4130
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