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Theorem sspwtr 32708
Description: Virtual deduction proof of the right-to-left implication of dftr4 4545. A class which is a subclass of its power class is transitive. This proof corresponds to the virtual deduction proof of sspwtr 32708 without accumulating results. (Contributed by Alan Sare, 2-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sspwtr  |-  ( A 
C_  ~P A  ->  Tr  A )

Proof of Theorem sspwtr
Dummy variables  z 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dftr2 4542 . . 3  |-  ( Tr  A  <->  A. z A. y
( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )
2 idn1 32440 . . . . . . . 8  |-  (. A  C_ 
~P A  ->.  A  C_  ~P A ).
3 idn2 32488 . . . . . . . . 9  |-  (. A  C_ 
~P A ,. (
z  e.  y  /\  y  e.  A )  ->.  ( z  e.  y  /\  y  e.  A ) ).
4 simpr 461 . . . . . . . . 9  |-  ( ( z  e.  y  /\  y  e.  A )  ->  y  e.  A )
53, 4e2 32506 . . . . . . . 8  |-  (. A  C_ 
~P A ,. (
z  e.  y  /\  y  e.  A )  ->.  y  e.  A ).
6 ssel 3498 . . . . . . . 8  |-  ( A 
C_  ~P A  ->  (
y  e.  A  -> 
y  e.  ~P A
) )
72, 5, 6e12 32610 . . . . . . 7  |-  (. A  C_ 
~P A ,. (
z  e.  y  /\  y  e.  A )  ->.  y  e.  ~P A ).
8 elpwi 4019 . . . . . . 7  |-  ( y  e.  ~P A  -> 
y  C_  A )
97, 8e2 32506 . . . . . 6  |-  (. A  C_ 
~P A ,. (
z  e.  y  /\  y  e.  A )  ->.  y 
C_  A ).
10 simpl 457 . . . . . . 7  |-  ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  y )
113, 10e2 32506 . . . . . 6  |-  (. A  C_ 
~P A ,. (
z  e.  y  /\  y  e.  A )  ->.  z  e.  y ).
12 ssel 3498 . . . . . 6  |-  ( y 
C_  A  ->  (
z  e.  y  -> 
z  e.  A ) )
139, 11, 12e22 32546 . . . . 5  |-  (. A  C_ 
~P A ,. (
z  e.  y  /\  y  e.  A )  ->.  z  e.  A ).
1413in2 32480 . . . 4  |-  (. A  C_ 
~P A  ->.  ( (
z  e.  y  /\  y  e.  A )  ->  z  e.  A ) ).
1514gen12 32493 . . 3  |-  (. A  C_ 
~P A  ->.  A. z A. y ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) ).
16 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 ) )
171, 15, 16e01 32566 . 2  |-  (. A  C_ 
~P A  ->.  Tr  A ).
1817in1 32437 1  |-  ( A 
C_  ~P A  ->  Tr  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1377    e. wcel 1767    C_ wss 3476   ~Pcpw 4010   Tr wtr 4540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115  df-in 3483  df-ss 3490  df-pw 4012  df-uni 4246  df-tr 4541  df-vd1 32436  df-vd2 32444
This theorem is referenced by: (None)
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