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Theorem sspwtrALT 37210
Description: Virtual deduction proof of sspwtr 37209. This proof is the same as the proof of sspwtr 37209 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 4499 . . 3  |-  ( Tr  A  <->  A. z A. y
( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )
2 simpr 463 . . . . . 6  |-  ( ( z  e.  y  /\  y  e.  A )  ->  y  e.  A )
3 ssel 3426 . . . . . 6  |-  ( A 
C_  ~P A  ->  (
y  e.  A  -> 
y  e.  ~P A
) )
4 elpwi 3960 . . . . . 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 459 . . . . . 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 3426 . . . . 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 1774 . . 3  |-  ( A 
C_  ~P A  ->  A. z A. y ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )
12 biimpr 202 . . 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 36832 1  |-  ( A 
C_  ~P A  ->  Tr  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371   A.wal 1442    e. wcel 1887    C_ wss 3404   ~Pcpw 3951   Tr wtr 4497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-v 3047  df-in 3411  df-ss 3418  df-pw 3953  df-uni 4199  df-tr 4498
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator