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Theorem sspwimpVD 28740
Description: The following User's Proof is a Virtual Deduction proof ( see: wvd1 28369) using conjunction-form virtual hypothesis collections. It was completed manually, but has the potential to be completed automatically by a tools program which would invoke Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sspwimp 28739 is sspwimpVD 28740 without virtual deductions and was derived from sspwimpVD 28740. (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 2919 . . . . . . 7  |-  x  e. 
_V
21vd01 28407 . . . . . 6  |-  (.  T.  ->.  x  e.  _V ).
3 idn1 28374 . . . . . . 7  |-  (. A  C_  B  ->.  A  C_  B ).
4 idn1 28374 . . . . . . . 8  |-  (. x  e.  ~P A  ->.  x  e.  ~P A ).
5 elpwi 3767 . . . . . . . 8  |-  ( x  e.  ~P A  ->  x  C_  A )
64, 5el1 28438 . . . . . . 7  |-  (. x  e.  ~P A  ->.  x  C_  A ).
7 sstr 3316 . . . . . . . 8  |-  ( ( x  C_  A  /\  A  C_  B )  ->  x  C_  B )
87ancoms 440 . . . . . . 7  |-  ( ( A  C_  B  /\  x  C_  A )  ->  x  C_  B )
93, 6, 8el12 28547 . . . . . 6  |-  (. (. A  C_  B ,. x  e.  ~P A ).  ->.  x  C_  B ).
102, 9elpwgdedVD 28738 . . . . . 6  |-  (. (.  T.  ,. (. A  C_  B ,. x  e.  ~P A ). ).  ->.  x  e.  ~P B ).
112, 9, 10un0.1 28600 . . . . 5  |-  (. (. A  C_  B ,. x  e.  ~P A ).  ->.  x  e.  ~P B ).
1211int2 28416 . . . 4  |-  (. A  C_  B  ->.  ( x  e. 
~P A  ->  x  e.  ~P B ) ).
1312gen11 28426 . . 3  |-  (. A  C_  B  ->.  A. x ( x  e.  ~P A  ->  x  e.  ~P B
) ).
14 dfss2 3297 . . . 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 28438 . 2  |-  (. A  C_  B  ->.  ~P A  C_  ~P B ).
1716in1 28371 1  |-  ( A 
C_  B  ->  ~P A  C_  ~P B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    T. wtru 1322   A.wal 1546    e. wcel 1721   _Vcvv 2916    C_ wss 3280   ~Pcpw 3759   (.wvhc2 28381
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918  df-in 3287  df-ss 3294  df-pw 3761  df-vd1 28370  df-vhc2 28382
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