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Theorem sspwtrALT 37071
Description: Virtual deduction proof of sspwtr 37070. This proof is the same as the proof of sspwtr 37070 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 4517 . . 3  |-  ( Tr  A  <->  A. z A. y
( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )
2 simpr 462 . . . . . 6  |-  ( ( z  e.  y  /\  y  e.  A )  ->  y  e.  A )
3 ssel 3458 . . . . . 6  |-  ( A 
C_  ~P A  ->  (
y  e.  A  -> 
y  e.  ~P A
) )
4 elpwi 3988 . . . . . 6  |-  ( y  e.  ~P A  -> 
y  C_  A )
52, 3, 4syl56 35 . . . . 5  |-  ( A 
C_  ~P A  ->  (
( z  e.  y  /\  y  e.  A
)  ->  y  C_  A ) )
6 idd 25 . . . . . 6  |-  ( A 
C_  ~P A  ->  (
( z  e.  y  /\  y  e.  A
)  ->  ( z  e.  y  /\  y  e.  A ) ) )
7 simpl 458 . . . . . 6  |-  ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  y )
86, 7syl6 34 . . . . 5  |-  ( A 
C_  ~P A  ->  (
( z  e.  y  /\  y  e.  A
)  ->  z  e.  y ) )
9 ssel 3458 . . . . 5  |-  ( y 
C_  A  ->  (
z  e.  y  -> 
z  e.  A ) )
105, 8, 9syl6c 66 . . . 4  |-  ( A 
C_  ~P A  ->  (
( z  e.  y  /\  y  e.  A
)  ->  z  e.  A ) )
1110alrimivv 1764 . . 3  |-  ( A 
C_  ~P A  ->  A. z A. y ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )
12 biimpr 201 . . 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 65 . 2  |-  ( A 
C_  ~P A  ->  Tr  A )
1413idiALT 36690 1  |-  ( A 
C_  ~P A  ->  Tr  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370   A.wal 1435    e. wcel 1868    C_ wss 3436   ~Pcpw 3979   Tr wtr 4515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-v 3083  df-in 3443  df-ss 3450  df-pw 3981  df-uni 4217  df-tr 4516
This theorem is referenced by: (None)
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