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Theorem sspwtr 31567
Description: Virtual deduction proof of the right-to-left implication of dftr4 4402. A class which is a subclass of its power class is transitive. This proof corresponds to the virtual deduction proof of sspwtr 31567 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 4399 . . 3  |-  ( Tr  A  <->  A. z A. y
( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )
2 idn1 31299 . . . . . . . 8  |-  (. A  C_ 
~P A  ->.  A  C_  ~P A ).
3 idn2 31347 . . . . . . . . 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 31365 . . . . . . . 8  |-  (. A  C_ 
~P A ,. (
z  e.  y  /\  y  e.  A )  ->.  y  e.  A ).
6 ssel 3362 . . . . . . . 8  |-  ( A 
C_  ~P A  ->  (
y  e.  A  -> 
y  e.  ~P A
) )
72, 5, 6e12 31469 . . . . . . 7  |-  (. A  C_ 
~P A ,. (
z  e.  y  /\  y  e.  A )  ->.  y  e.  ~P A ).
8 elpwi 3881 . . . . . . 7  |-  ( y  e.  ~P A  -> 
y  C_  A )
97, 8e2 31365 . . . . . 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 31365 . . . . . 6  |-  (. A  C_ 
~P A ,. (
z  e.  y  /\  y  e.  A )  ->.  z  e.  y ).
12 ssel 3362 . . . . . 6  |-  ( y 
C_  A  ->  (
z  e.  y  -> 
z  e.  A ) )
139, 11, 12e22 31405 . . . . 5  |-  (. A  C_ 
~P A ,. (
z  e.  y  /\  y  e.  A )  ->.  z  e.  A ).
1413in2 31339 . . . 4  |-  (. A  C_ 
~P A  ->.  ( (
z  e.  y  /\  y  e.  A )  ->  z  e.  A ) ).
1514gen12 31352 . . 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 31425 . 2  |-  (. A  C_ 
~P A  ->.  Tr  A ).
1817in1 31296 1  |-  ( A 
C_  ~P A  ->  Tr  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1367    e. wcel 1756    C_ wss 3340   ~Pcpw 3872   Tr wtr 4397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-v 2986  df-in 3347  df-ss 3354  df-pw 3874  df-uni 4104  df-tr 4398  df-vd1 31295  df-vd2 31303
This theorem is referenced by: (None)
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