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Theorem sspwimpcfVD 28374
Description: The following User's Proof is a Virtual Deduction proof ( see: wvd1 28001) 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 28373 is sspwimpcfVD 28374 without virtual deductions and was derived from sspwimpcfVD 28374. 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 2902 . . . . . 6  |-  x  e. 
_V
2 idn1 28006 . . . . . . 7  |-  (. A  C_  B  ->.  A  C_  B ).
3 idn1 28006 . . . . . . . 8  |-  (. x  e.  ~P A  ->.  x  e.  ~P A ).
4 elpwi 3750 . . . . . . . 8  |-  ( x  e.  ~P A  ->  x  C_  A )
53, 4el1 28070 . . . . . . 7  |-  (. x  e.  ~P A  ->.  x  C_  A ).
6 sstr2 3298 . . . . . . . 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 28179 . . . . . 6  |-  (. (. A  C_  B ,. x  e.  ~P A ).  ->.  x  C_  B ).
9 elpwg 3749 . . . . . . 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 28143 . . . . 5  |-  (. (. A  C_  B ,. x  e.  ~P A ).  ->.  x  e.  ~P B ).
1211int2 28048 . . . 4  |-  (. A  C_  B  ->.  ( x  e. 
~P A  ->  x  e.  ~P B ) ).
1312gen11 28058 . . 3  |-  (. A  C_  B  ->.  A. x ( x  e.  ~P A  ->  x  e.  ~P B
) ).
14 dfss2 3280 . . . 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 28070 . 2  |-  (. A  C_  B  ->.  ~P A  C_  ~P B ).
1716in1 28003 1  |-  ( A 
C_  B  ->  ~P A  C_  ~P B )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1546    e. wcel 1717   _Vcvv 2899    C_ wss 3263   ~Pcpw 3742
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-v 2901  df-in 3270  df-ss 3277  df-pw 3744  df-vd1 28002  df-vhc2 28014
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