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

Theorem ssunieq 4286
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 4281 . . 3  |-  ( A  e.  B  ->  A  C_ 
U. B )
2 unissb 4283 . . . 4  |-  ( U. B  C_  A  <->  A. x  e.  B  x  C_  A
)
32biimpri 206 . . 3  |-  ( A. x  e.  B  x  C_  A  ->  U. B  C_  A )
41, 3anim12i 566 . 2  |-  ( ( A  e.  B  /\  A. x  e.  B  x 
C_  A )  -> 
( A  C_  U. B  /\  U. B  C_  A
) )
5 eqss 3514 . 2  |-  ( A  =  U. B  <->  ( A  C_ 
U. B  /\  U. B  C_  A ) )
64, 5sylibr 212 1  |-  ( ( A  e.  B  /\  A. x  e.  B  x 
C_  A )  ->  A  =  U. B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807    C_ wss 3471   U.cuni 4251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-v 3111  df-in 3478  df-ss 3485  df-uni 4252
This theorem is referenced by:  unimax  4287  shsspwh  26290
  Copyright terms: Public domain W3C validator