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Theorem sspwtrALT 31858
Description: Virtual deduction proof of sspwtr 31857. This proof is the same as the proof of sspwtr 31857 except each virtual deduction symbol is replaced by its non-virtual deduction symbol equivalent. A class which is a subclass of its power class is transitive. (Contributed by Alan Sare, 3-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sspwtrALT  |-  ( A 
C_  ~P A  ->  Tr  A )

Proof of Theorem sspwtrALT
Dummy variables  z 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dftr2 4487 . . 3  |-  ( Tr  A  <->  A. z A. y
( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )
2 simpr 461 . . . . . 6  |-  ( ( z  e.  y  /\  y  e.  A )  ->  y  e.  A )
3 ssel 3450 . . . . . 6  |-  ( A 
C_  ~P A  ->  (
y  e.  A  -> 
y  e.  ~P A
) )
4 elpwi 3969 . . . . . 6  |-  ( y  e.  ~P A  -> 
y  C_  A )
52, 3, 4syl56 34 . . . . 5  |-  ( A 
C_  ~P A  ->  (
( z  e.  y  /\  y  e.  A
)  ->  y  C_  A ) )
6 idd 24 . . . . . 6  |-  ( A 
C_  ~P A  ->  (
( z  e.  y  /\  y  e.  A
)  ->  ( z  e.  y  /\  y  e.  A ) ) )
7 simpl 457 . . . . . 6  |-  ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  y )
86, 7syl6 33 . . . . 5  |-  ( A 
C_  ~P A  ->  (
( z  e.  y  /\  y  e.  A
)  ->  z  e.  y ) )
9 ssel 3450 . . . . 5  |-  ( y 
C_  A  ->  (
z  e.  y  -> 
z  e.  A ) )
105, 8, 9syl6c 64 . . . 4  |-  ( A 
C_  ~P A  ->  (
( z  e.  y  /\  y  e.  A
)  ->  z  e.  A ) )
1110alrimivv 1687 . . 3  |-  ( A 
C_  ~P A  ->  A. z A. y ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )
12 bi2 198 . . 3  |-  ( ( Tr  A  <->  A. z A. y ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )  -> 
( A. z A. y ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A )  ->  Tr  A ) )
131, 11, 12mpsyl 63 . 2  |-  ( A 
C_  ~P A  ->  Tr  A )
1413idiALT 31455 1  |-  ( A 
C_  ~P A  ->  Tr  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1368    e. wcel 1758    C_ wss 3428   ~Pcpw 3960   Tr wtr 4485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-v 3072  df-in 3435  df-ss 3442  df-pw 3962  df-uni 4192  df-tr 4486
This theorem is referenced by: (None)
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