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Theorem sspwtrALT2 28873
Description: Short predicate calculus proof of the right-to-left implication of dftr4 4299. 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 28872, which is the virtual deduction proof sspwtr 28871 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 3334 . . . . . 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 3799 . . . . 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 3334 . . . 4  |-  ( y 
C_  A  ->  (
z  e.  y  -> 
z  e.  A ) )
84, 6, 7ee22 1371 . . 3  |-  ( A 
C_  ~P A  ->  (
( z  e.  y  /\  y  e.  A
)  ->  z  e.  A ) )
98alrimivv 1642 . 2  |-  ( A 
C_  ~P A  ->  A. z A. y ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )
10 dftr2 4296 . 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 1549    e. wcel 1725    C_ wss 3312   ~Pcpw 3791   Tr wtr 4294
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-in 3319  df-ss 3326  df-pw 3793  df-uni 4008  df-tr 4295
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