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

Theorem unissel 3638
Description: Condition turning a subclass relationship for union into an equality. (Contributed by NM, 18-Jul-2006.)
Assertion
Ref Expression
unissel  |-  ( ( U. A  C_  B  /\  B  e.  A
)  ->  U. A  =  B )

Proof of Theorem unissel
StepHypRef Expression
1 simpl 107 . 2  |-  ( ( U. A  C_  B  /\  B  e.  A
)  ->  U. A  C_  B )
2 elssuni 3637 . . 3  |-  ( B  e.  A  ->  B  C_ 
U. A )
32adantl 271 . 2  |-  ( ( U. A  C_  B  /\  B  e.  A
)  ->  B  C_  U. A
)
41, 3eqssd 3017 1  |-  ( ( U. A  C_  B  /\  B  e.  A
)  ->  U. A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434    C_ wss 2974   U.cuni 3609
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-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 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-in 2980  df-ss 2987  df-uni 3610
This theorem is referenced by:  elpwuni  3770
  Copyright terms: Public domain W3C validator