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Theorem sspwimpVD 29105
Description: The following User's Proof is a Virtual Deduction proof ( see: wvd1 28734) 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 29104 is sspwimpVD 29105 without virtual deductions and was derived from sspwimpVD 29105. (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 2961 . . . . . . 7  |-  x  e. 
_V
21vd01 28772 . . . . . 6  |-  (.  T.  ->.  x  e.  _V ).
3 idn1 28739 . . . . . . 7  |-  (. A  C_  B  ->.  A  C_  B ).
4 idn1 28739 . . . . . . . 8  |-  (. x  e.  ~P A  ->.  x  e.  ~P A ).
5 elpwi 3809 . . . . . . . 8  |-  ( x  e.  ~P A  ->  x  C_  A )
64, 5el1 28803 . . . . . . 7  |-  (. x  e.  ~P A  ->.  x  C_  A ).
7 sstr 3358 . . . . . . . 8  |-  ( ( x  C_  A  /\  A  C_  B )  ->  x  C_  B )
87ancoms 441 . . . . . . 7  |-  ( ( A  C_  B  /\  x  C_  A )  ->  x  C_  B )
93, 6, 8el12 28912 . . . . . 6  |-  (. (. A  C_  B ,. x  e.  ~P A ).  ->.  x  C_  B ).
102, 9elpwgdedVD 29103 . . . . . 6  |-  (. (.  T.  ,. (. A  C_  B ,. x  e.  ~P A ). ).  ->.  x  e.  ~P B ).
112, 9, 10un0.1 28965 . . . . 5  |-  (. (. A  C_  B ,. x  e.  ~P A ).  ->.  x  e.  ~P B ).
1211int2 28781 . . . 4  |-  (. A  C_  B  ->.  ( x  e. 
~P A  ->  x  e.  ~P B ) ).
1312gen11 28791 . . 3  |-  (. A  C_  B  ->.  A. x ( x  e.  ~P A  ->  x  e.  ~P B
) ).
14 dfss2 3339 . . . 4  |-  ( ~P A  C_  ~P B  <->  A. x ( x  e. 
~P A  ->  x  e.  ~P B ) )
1514biimpri 199 . . 3  |-  ( A. x ( x  e. 
~P A  ->  x  e.  ~P B )  ->  ~P A  C_  ~P B
)
1613, 15el1 28803 . 2  |-  (. A  C_  B  ->.  ~P A  C_  ~P B ).
1716in1 28736 1  |-  ( A 
C_  B  ->  ~P A  C_  ~P B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    T. wtru 1326   A.wal 1550    e. wcel 1726   _Vcvv 2958    C_ wss 3322   ~Pcpw 3801   (.wvhc2 28746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-in 3329  df-ss 3336  df-pw 3803  df-vd1 28735  df-vhc2 28747
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