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Theorem sspwimpVD 28695
Description: The following User's Proof is a Virtual Deduction proof ( see: wvd1 28337) using conjunction-form virtual hypothesis collections. It was completed manually, but has the potential to be completed automatically by a tools program which would invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sspwimp 28694 is sspwimpVD 28695 without virtual deductions and was derived from sspwimpVD 28695. (Contributed by Alan Sare, 23-Apr-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
sspwimpVD  |-  ( A 
C_  B  ->  ~P A  C_  ~P B )

Proof of Theorem sspwimpVD
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 vex 2791 . . . . . . 7  |-  x  e. 
_V
21vd01 28369 . . . . . 6  |-  (.  T.  ->.  x  e.  _V ).
3 idn1 28342 . . . . . . 7  |-  (. A  C_  B  ->.  A  C_  B ).
4 idn1 28342 . . . . . . . 8  |-  (. x  e.  ~P A  ->.  x  e.  ~P A ).
5 elpwi 3633 . . . . . . . 8  |-  ( x  e.  ~P A  ->  x  C_  A )
64, 5el1 28400 . . . . . . 7  |-  (. x  e.  ~P A  ->.  x  C_  A ).
7 sstr 3187 . . . . . . . 8  |-  ( ( x  C_  A  /\  A  C_  B )  ->  x  C_  B )
87ancoms 439 . . . . . . 7  |-  ( ( A  C_  B  /\  x  C_  A )  ->  x  C_  B )
93, 6, 8el12 28501 . . . . . 6  |-  (. (. A  C_  B ,. x  e.  ~P A ).  ->.  x  C_  B ).
102, 9elpwgdedVD 28693 . . . . . 6  |-  (. (.  T.  ,. (. A  C_  B ,. x  e.  ~P A ). ).  ->.  x  e.  ~P B ).
112, 9, 10un0.1 28554 . . . . 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    T. wtru 1307   A.wal 1527    e. wcel 1684   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625   (.wvhc2 28349
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
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