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Theorem sspwtrALT2 28279
Description: Short predicate calculus proof of the right-to-left implication of dftr4 4250. 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 28278, which is the virtual deduction proof sspwtr 28277 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 3287 . . . . . 6  |-  ( A 
C_  ~P A  ->  (
y  e.  A  -> 
y  e.  ~P A
) )
21adantld 454 . . . . 5  |-  ( A 
C_  ~P A  ->  (
( z  e.  y  /\  y  e.  A
)  ->  y  e.  ~P A ) )
3 elpwi 3752 . . . . 5  |-  ( y  e.  ~P A  -> 
y  C_  A )
42, 3syl6 31 . . . 4  |-  ( A 
C_  ~P A  ->  (
( z  e.  y  /\  y  e.  A
)  ->  y  C_  A ) )
5 simpl 444 . . . . 5  |-  ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  y )
65a1i 11 . . . 4  |-  ( A 
C_  ~P A  ->  (
( z  e.  y  /\  y  e.  A
)  ->  z  e.  y ) )
7 ssel 3287 . . . 4  |-  ( y 
C_  A  ->  (
z  e.  y  -> 
z  e.  A ) )
84, 6, 7ee22 1368 . . 3  |-  ( A 
C_  ~P A  ->  (
( z  e.  y  /\  y  e.  A
)  ->  z  e.  A ) )
98alrimivv 1639 . 2  |-  ( A 
C_  ~P A  ->  A. z A. y ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )
10 dftr2 4247 . 2  |-  ( Tr  A  <->  A. z A. y
( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )
119, 10sylibr 204 1  |-  ( A 
C_  ~P A  ->  Tr  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1546    e. wcel 1717    C_ wss 3265   ~Pcpw 3744   Tr wtr 4245
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-v 2903  df-in 3272  df-ss 3279  df-pw 3746  df-uni 3960  df-tr 4246
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