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

Theorem sspwimpcfVD 28697
Description: The following User's Proof is a Virtual Deduction proof ( see: wvd1 28337) using conjunction-form virtual hypothesis collections. It was completed automatically by a tools program which would invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sspwimpcf 28696 is sspwimpcfVD 28697 without virtual deductions and was derived from sspwimpcfVD 28697. The version of completeusersproof.cmd used is capable of only generating conjunction-form unification theorems, not unification deductions. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
1::  |-  (. A  C_  B  ->.  A  C_  B ).
2::  |-  (. ...........  x  e.  ~P A  ->.  x  e.  ~P A ).
3:2:  |-  (. ...........  x  e.  ~P A  ->.  x  C_  A ).
4:3,1:  |-  (. (. A  C_  B ,. x  e.  ~P A ).  ->.  x  C_  B ).
5::  |-  x  e.  _V
6:4,5:  |-  (. (. A  C_  B ,. x  e.  ~P A ).  ->.  x  e.  ~P B  ).
7:6:  |-  (. A  C_  B  ->.  ( x  e.  ~P A  ->  x  e.  ~P B )  ).
8:7:  |-  (. A  C_  B  ->.  A. x ( x  e.  ~P A  ->  x  e.  ~P B ) ).
9:8:  |-  (. A  C_  B  ->.  ~P A  C_  ~P B ).
qed:9:  |-  ( A  C_  B  ->  ~P A  C_  ~P B )
Assertion
Ref Expression
sspwimpcfVD  |-  ( A 
C_  B  ->  ~P A  C_  ~P B )

Proof of Theorem sspwimpcfVD
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 vex 2791 . . . . . 6  |-  x  e. 
_V
2 idn1 28342 . . . . . . 7  |-  (. A  C_  B  ->.  A  C_  B ).
3 idn1 28342 . . . . . . . 8  |-  (. x  e.  ~P A  ->.  x  e.  ~P A ).
4 elpwi 3633 . . . . . . . 8  |-  ( x  e.  ~P A  ->  x  C_  A )
53, 4el1 28400 . . . . . . 7  |-  (. x  e.  ~P A  ->.  x  C_  A ).
6 sstr2 3186 . . . . . . . 8  |-  ( x 
C_  A  ->  ( A  C_  B  ->  x  C_  B ) )
76impcom 419 . . . . . . 7  |-  ( ( A  C_  B  /\  x  C_  A )  ->  x  C_  B )
82, 5, 7el12 28501 . . . . . 6  |-  (. (. A  C_  B ,. x  e.  ~P A ).  ->.  x  C_  B ).
9 elpwg 3632 . . . . . . 7  |-  ( x  e.  _V  ->  (
x  e.  ~P B  <->  x 
C_  B ) )
109biimpar 471 . . . . . 6  |-  ( ( x  e.  _V  /\  x  C_  B )  ->  x  e.  ~P B
)
111, 8, 10el021old 28474 . . . . 5  |-  (. (. A  C_  B ,. x  e.  ~P A ).  ->.  x  e.  ~P B ).
1211int2 28378 . . . 4  |-  (. A  C_  B  ->.  ( x  e. 
~P A  ->  x  e.  ~P B ) ).
1312gen11 28388 . . 3  |-  (. A  C_  B  ->.  A. x ( x  e.  ~P A  ->  x  e.  ~P B
) ).
14 dfss2 3169 . . . 4  |-  ( ~P A  C_  ~P B  <->  A. x ( x  e. 
~P A  ->  x  e.  ~P B ) )
1514biimpri 197 . . 3  |-  ( A. x ( x  e. 
~P A  ->  x  e.  ~P B )  ->  ~P A  C_  ~P B
)
1613, 15el1 28400 . 2  |-  (. A  C_  B  ->.  ~P A  C_  ~P B ).
1716in1 28339 1  |-  ( A 
C_  B  ->  ~P A  C_  ~P B )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1527    e. wcel 1684   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-in 3159  df-ss 3166  df-pw 3627  df-vd1 28338  df-vhc2 28350
  Copyright terms: Public domain W3C validator