ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ssequn1 Unicode version

Theorem ssequn1 3193
Description: A relationship between subclass and union. Theorem 26 of [Suppes] p. 27. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
ssequn1  |-  ( A 
C_  B  <->  ( A  u.  B )  =  B )

Proof of Theorem ssequn1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 bicom 139 . . . 4  |-  ( ( x  e.  B  <->  ( x  e.  A  \/  x  e.  B ) )  <->  ( (
x  e.  A  \/  x  e.  B )  <->  x  e.  B ) )
2 pm4.72 778 . . . 4  |-  ( ( x  e.  A  ->  x  e.  B )  <->  ( x  e.  B  <->  ( x  e.  A  \/  x  e.  B ) ) )
3 elun 3164 . . . . 5  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
43bibi1i 227 . . . 4  |-  ( ( x  e.  ( A  u.  B )  <->  x  e.  B )  <->  ( (
x  e.  A  \/  x  e.  B )  <->  x  e.  B ) )
51, 2, 43bitr4i 211 . . 3  |-  ( ( x  e.  A  ->  x  e.  B )  <->  ( x  e.  ( A  u.  B )  <->  x  e.  B ) )
65albii 1414 . 2  |-  ( A. x ( x  e.  A  ->  x  e.  B )  <->  A. x
( x  e.  ( A  u.  B )  <-> 
x  e.  B ) )
7 dfss2 3036 . 2  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
8 dfcleq 2094 . 2  |-  ( ( A  u.  B )  =  B  <->  A. x
( x  e.  ( A  u.  B )  <-> 
x  e.  B ) )
96, 7, 83bitr4i 211 1  |-  ( A 
C_  B  <->  ( A  u.  B )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    \/ wo 670   A.wal 1297    = wceq 1299    e. wcel 1448    u. cun 3019    C_ wss 3021
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-un 3025  df-in 3027  df-ss 3034
This theorem is referenced by:  ssequn2  3196  uniop  4115  pwssunim  4144  unisuc  4273  unisucg  4274  rdgisucinc  6212  oasuc  6290  omsuc  6298  undifdc  6741
  Copyright terms: Public domain W3C validator