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Theorem sspwtrALT2 33969
Description: Short predicate calculus proof of the right-to-left implication of dftr4 4465. 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 33960, which is the virtual deduction proof sspwtr 33959 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 3411 . . . . . 6  |-  ( A 
C_  ~P A  ->  (
y  e.  A  -> 
y  e.  ~P A
) )
21adantld 465 . . . . 5  |-  ( A 
C_  ~P A  ->  (
( z  e.  y  /\  y  e.  A
)  ->  y  e.  ~P A ) )
3 elpwi 3936 . . . . 5  |-  ( y  e.  ~P A  -> 
y  C_  A )
42, 3syl6 33 . . . 4  |-  ( A 
C_  ~P A  ->  (
( z  e.  y  /\  y  e.  A
)  ->  y  C_  A ) )
5 simpl 455 . . . . 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 3411 . . . 4  |-  ( y 
C_  A  ->  (
z  e.  y  -> 
z  e.  A ) )
84, 6, 7syl6c 64 . . 3  |-  ( A 
C_  ~P A  ->  (
( z  e.  y  /\  y  e.  A
)  ->  z  e.  A ) )
98alrimivv 1728 . 2  |-  ( A 
C_  ~P A  ->  A. z A. y ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )
10 dftr2 4462 . 2  |-  ( Tr  A  <->  A. z A. y
( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )
119, 10sylibr 212 1  |-  ( A 
C_  ~P A  ->  Tr  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367   A.wal 1397    e. wcel 1826    C_ wss 3389   ~Pcpw 3927   Tr wtr 4460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-v 3036  df-in 3396  df-ss 3403  df-pw 3929  df-uni 4164  df-tr 4461
This theorem is referenced by: (None)
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