Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sspwtr Structured version   Visualization version   Unicode version

Theorem sspwtr 37249
Description: Virtual deduction proof of the right-to-left implication of dftr4 4516. A class which is a subclass of its power class is transitive. This proof corresponds to the virtual deduction proof of sspwtr 37249 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 4513 . . 3  |-  ( Tr  A  <->  A. z A. y
( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )
2 idn1 36987 . . . . . . . 8  |-  (. A  C_ 
~P A  ->.  A  C_  ~P A ).
3 idn2 37035 . . . . . . . . 9  |-  (. A  C_ 
~P A ,. (
z  e.  y  /\  y  e.  A )  ->.  ( z  e.  y  /\  y  e.  A ) ).
4 simpr 467 . . . . . . . . 9  |-  ( ( z  e.  y  /\  y  e.  A )  ->  y  e.  A )
53, 4e2 37053 . . . . . . . 8  |-  (. A  C_ 
~P A ,. (
z  e.  y  /\  y  e.  A )  ->.  y  e.  A ).
6 ssel 3438 . . . . . . . 8  |-  ( A 
C_  ~P A  ->  (
y  e.  A  -> 
y  e.  ~P A
) )
72, 5, 6e12 37151 . . . . . . 7  |-  (. A  C_ 
~P A ,. (
z  e.  y  /\  y  e.  A )  ->.  y  e.  ~P A ).
8 elpwi 3972 . . . . . . 7  |-  ( y  e.  ~P A  -> 
y  C_  A )
97, 8e2 37053 . . . . . 6  |-  (. A  C_ 
~P A ,. (
z  e.  y  /\  y  e.  A )  ->.  y 
C_  A ).
10 simpl 463 . . . . . . 7  |-  ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  y )
113, 10e2 37053 . . . . . 6  |-  (. A  C_ 
~P A ,. (
z  e.  y  /\  y  e.  A )  ->.  z  e.  y ).
12 ssel 3438 . . . . . 6  |-  ( y 
C_  A  ->  (
z  e.  y  -> 
z  e.  A ) )
139, 11, 12e22 37093 . . . . 5  |-  (. A  C_ 
~P A ,. (
z  e.  y  /\  y  e.  A )  ->.  z  e.  A ).
1413in2 37027 . . . 4  |-  (. A  C_ 
~P A  ->.  ( (
z  e.  y  /\  y  e.  A )  ->  z  e.  A ) ).
1514gen12 37040 . . 3  |-  (. A  C_ 
~P A  ->.  A. z A. y ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) ).
16 biimpr 203 . . 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 37113 . 2  |-  (. A  C_ 
~P A  ->.  Tr  A ).
1817in1 36984 1  |-  ( A 
C_  ~P A  ->  Tr  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375   A.wal 1453    e. wcel 1898    C_ wss 3416   ~Pcpw 3963   Tr wtr 4511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-v 3059  df-in 3423  df-ss 3430  df-pw 3965  df-uni 4213  df-tr 4512  df-vd1 36983  df-vd2 36991
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator