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

Theorem sspwimpcfVD 28970
Description: The following User's Proof is a Virtual Deduction proof ( see: wvd1 28597) 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 28969 is sspwimpcfVD 28970 without virtual deductions and was derived from sspwimpcfVD 28970. 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 2951 . . . . . 6  |-  x  e. 
_V
2 idn1 28602 . . . . . . 7  |-  (. A  C_  B  ->.  A  C_  B ).
3 idn1 28602 . . . . . . . 8  |-  (. x  e.  ~P A  ->.  x  e.  ~P A ).
4 elpwi 3799 . . . . . . . 8  |-  ( x  e.  ~P A  ->  x  C_  A )
53, 4el1 28666 . . . . . . 7  |-  (. x  e.  ~P A  ->.  x  C_  A ).
6 sstr2 3347 . . . . . . . 8  |-  ( x 
C_  A  ->  ( A  C_  B  ->  x  C_  B ) )
76impcom 420 . . . . . . 7  |-  ( ( A  C_  B  /\  x  C_  A )  ->  x  C_  B )
82, 5, 7el12 28775 . . . . . 6  |-  (. (. A  C_  B ,. x  e.  ~P A ).  ->.  x  C_  B ).
9 elpwg 3798 . . . . . . 7  |-  ( x  e.  _V  ->  (
x  e.  ~P B  <->  x 
C_  B ) )
109biimpar 472 . . . . . 6  |-  ( ( x  e.  _V  /\  x  C_  B )  ->  x  e.  ~P B
)
111, 8, 10el021old 28739 . . . . 5  |-  (. (. A  C_  B ,. x  e.  ~P A ).  ->.  x  e.  ~P B ).
1211int2 28644 . . . 4  |-  (. A  C_  B  ->.  ( x  e. 
~P A  ->  x  e.  ~P B ) ).
1312gen11 28654 . . 3  |-  (. A  C_  B  ->.  A. x ( x  e.  ~P A  ->  x  e.  ~P B
) ).
14 dfss2 3329 . . . 4  |-  ( ~P A  C_  ~P B  <->  A. x ( x  e. 
~P A  ->  x  e.  ~P B ) )
1514biimpri 198 . . 3  |-  ( A. x ( x  e. 
~P A  ->  x  e.  ~P B )  ->  ~P A  C_  ~P B
)
1613, 15el1 28666 . 2  |-  (. A  C_  B  ->.  ~P A  C_  ~P B ).
1716in1 28599 1  |-  ( A 
C_  B  ->  ~P A  C_  ~P B )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1549    e. wcel 1725   _Vcvv 2948    C_ wss 3312   ~Pcpw 3791
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-in 3319  df-ss 3326  df-pw 3793  df-vd1 28598  df-vhc2 28610
  Copyright terms: Public domain W3C validator