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

Theorem disjpss 3272
Description: A class is a proper subset of its union with a disjoint nonempty class. (Contributed by NM, 15-Sep-2004.)
Assertion
Ref Expression
disjpss  i^i  (/)  =/=  (/)  C.  u.

Proof of Theorem disjpss
StepHypRef Expression
1 ssid 2958 . . . . . . . 8  C_
21biantru 286 . . . . . . 7 
C_ 
C_  C_
3 ssin 3153 . . . . . . 7  C_  C_  C_  i^i
42, 3bitri 173 . . . . . 6 
C_  C_  i^i
5 sseq2 2961 . . . . . 6  i^i  (/)  C_  i^i  C_  (/)
64, 5syl5bb 181 . . . . 5  i^i  (/)  C_  C_  (/)
7 ss0 3251 . . . . 5 
C_  (/)  (/)
86, 7syl6bi 152 . . . 4  i^i  (/)  C_  (/)
98necon3ad 2241 . . 3  i^i  (/)  =/=  (/)  C_
109imp 115 . 2  i^i  (/)  =/=  (/)  C_
11 nsspssun 3164 . . 3  C_  C.  u.
12 uncom 3081 . . . 4  u.  u.
1312psseq2i 3028 . . 3  C.  u.  C.  u.
1411, 13bitri 173 . 2  C_  C.  u.
1510, 14sylib 127 1  i^i  (/)  =/=  (/)  C.  u.
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wa 97   wceq 1242    =/= wne 2201    u. cun 2909    i^i cin 2910    C_ wss 2911    C. wpss 2912   (/)c0 3218
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-in1 544  ax-in2 545  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-ne 2203  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pss 2927  df-nul 3219
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator