MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssunieq Unicode version

Theorem ssunieq 3992
Description: Relationship implying union. (Contributed by NM, 10-Nov-1999.)
Assertion
Ref Expression
ssunieq  |-  ( ( A  e.  B  /\  A. x  e.  B  x 
C_  A )  ->  A  =  U. B )
Distinct variable groups:    x, A    x, B

Proof of Theorem ssunieq
StepHypRef Expression
1 elssuni 3987 . . 3  |-  ( A  e.  B  ->  A  C_ 
U. B )
2 unissb 3989 . . . 4  |-  ( U. B  C_  A  <->  A. x  e.  B  x  C_  A
)
32biimpri 198 . . 3  |-  ( A. x  e.  B  x  C_  A  ->  U. B  C_  A )
41, 3anim12i 550 . 2  |-  ( ( A  e.  B  /\  A. x  e.  B  x 
C_  A )  -> 
( A  C_  U. B  /\  U. B  C_  A
) )
5 eqss 3308 . 2  |-  ( A  =  U. B  <->  ( A  C_ 
U. B  /\  U. B  C_  A ) )
64, 5sylibr 204 1  |-  ( ( A  e.  B  /\  A. x  e.  B  x 
C_  A )  ->  A  =  U. B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2651    C_ wss 3265   U.cuni 3959
This theorem is referenced by:  unimax  3993  shsspwh  22598
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ral 2656  df-v 2903  df-in 3272  df-ss 3279  df-uni 3960
  Copyright terms: Public domain W3C validator