ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  pwssunim Unicode version

Theorem pwssunim 4012
Description: The power class of the union of two classes is a subset of the union of their power classes, if one class is a subclass of the other. One direction of Exercise 4.12(l) of [Mendelson] p. 235. (Contributed by Jim Kingdon, 30-Sep-2018.)
Assertion
Ref Expression
pwssunim  C_  C_ 
~P  u.  C_  ~P  u.  ~P

Proof of Theorem pwssunim
StepHypRef Expression
1 ssequn2 3110 . . . . 5 
C_  u.
2 pweq 3354 . . . . . 6  u.  ~P  u.  ~P
3 eqimss 2991 . . . . . 6  ~P  u.  ~P  ~P  u.  C_  ~P
42, 3syl 14 . . . . 5  u.  ~P  u.  C_  ~P
51, 4sylbi 114 . . . 4 
C_  ~P  u.  C_  ~P
6 ssequn1 3107 . . . . 5 
C_  u.
7 pweq 3354 . . . . . 6  u.  ~P  u.  ~P
8 eqimss 2991 . . . . . 6  ~P  u.  ~P  ~P  u.  C_  ~P
97, 8syl 14 . . . . 5  u.  ~P  u.  C_  ~P
106, 9sylbi 114 . . . 4 
C_  ~P  u.  C_  ~P
115, 10orim12i 675 . . 3  C_  C_  ~P  u.  C_  ~P  ~P  u.  C_  ~P
1211orcoms 648 . 2  C_  C_  ~P  u.  C_  ~P  ~P  u.  C_  ~P
13 ssun 3116 . 2  ~P  u.  C_  ~P  ~P  u.  C_  ~P  ~P  u.  C_  ~P  u.  ~P
1412, 13syl 14 1  C_  C_ 
~P  u.  C_  ~P  u.  ~P
Colors of variables: wff set class
Syntax hints:   wi 4   wo 628   wceq 1242    u. cun 2909    C_ wss 2911   ~Pcpw 3351
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353
This theorem is referenced by:  pwunim  4014
  Copyright terms: Public domain W3C validator