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

Theorem unidif 3653
Description: If the difference  A  \  B contains the largest members of  A, then the union of the difference is the union of  A. (Contributed by NM, 22-Mar-2004.)
Assertion
Ref Expression
unidif  |-  ( A. x  e.  A  E. y  e.  ( A  \  B ) x  C_  y  ->  U. ( A  \  B )  =  U. A )
Distinct variable groups:    x, y, A   
x, B, y

Proof of Theorem unidif
StepHypRef Expression
1 uniss2 3652 . . 3  |-  ( A. x  e.  A  E. y  e.  ( A  \  B ) x  C_  y  ->  U. A  C_  U. ( A  \  B ) )
2 difss 3108 . . . 4  |-  ( A 
\  B )  C_  A
32unissi 3644 . . 3  |-  U. ( A  \  B )  C_  U. A
41, 3jctil 305 . 2  |-  ( A. x  e.  A  E. y  e.  ( A  \  B ) x  C_  y  ->  ( U. ( A  \  B )  C_  U. A  /\  U. A  C_ 
U. ( A  \  B ) ) )
5 eqss 3023 . 2  |-  ( U. ( A  \  B )  =  U. A  <->  ( U. ( A  \  B ) 
C_  U. A  /\  U. A  C_  U. ( A 
\  B ) ) )
64, 5sylibr 132 1  |-  ( A. x  e.  A  E. y  e.  ( A  \  B ) x  C_  y  ->  U. ( A  \  B )  =  U. A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285   A.wral 2353   E.wrex 2354    \ cdif 2979    C_ wss 2982   U.cuni 3621
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-dif 2984  df-in 2988  df-ss 2995  df-uni 3622
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator