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

Theorem unissb 3601
Description: Relationship involving membership, subset, and union. Exercise 5 of [Enderton] p. 26 and its converse. (Contributed by NM, 20-Sep-2003.)
Assertion
Ref Expression
unissb  U.  C_  C_
Distinct variable groups:   ,   ,

Proof of Theorem unissb
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eluni 3574 . . . . . 6  U.
21imbi1i 227 . . . . 5  U.
3 19.23v 1760 . . . . 5
42, 3bitr4i 176 . . . 4  U.
54albii 1356 . . 3  U.
6 alcom 1364 . . . 4
7 19.21v 1750 . . . . . 6
8 impexp 250 . . . . . . . 8
9 bi2.04 237 . . . . . . . 8
108, 9bitri 173 . . . . . . 7
1110albii 1356 . . . . . 6
12 dfss2 2928 . . . . . . 7 
C_
1312imbi2i 215 . . . . . 6 
C_
147, 11, 133bitr4i 201 . . . . 5  C_
1514albii 1356 . . . 4  C_
166, 15bitri 173 . . 3  C_
175, 16bitri 173 . 2  U.  C_
18 dfss2 2928 . 2  U.  C_  U.
19 df-ral 2305 . 2  C_ 
C_
2017, 18, 193bitr4i 201 1  U.  C_  C_
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98  wal 1240  wex 1378   wcel 1390  wral 2300    C_ wss 2911   U.cuni 3571
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-bnd 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-ral 2305  df-v 2553  df-in 2918  df-ss 2925  df-uni 3572
This theorem is referenced by:  uniss2  3602  ssunieq  3604  sspwuni  3730  pwssb  3731  bm2.5ii  4188
  Copyright terms: Public domain W3C validator