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Theorem ssunieq 4246
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 4241 . . 3  |-  ( A  e.  B  ->  A  C_ 
U. B )
2 unissb 4243 . . . 4  |-  ( U. B  C_  A  <->  A. x  e.  B  x  C_  A
)
32biimpri 211 . . 3  |-  ( A. x  e.  B  x  C_  A  ->  U. B  C_  A )
41, 3anim12i 574 . 2  |-  ( ( A  e.  B  /\  A. x  e.  B  x 
C_  A )  -> 
( A  C_  U. B  /\  U. B  C_  A
) )
5 eqss 3459 . 2  |-  ( A  =  U. B  <->  ( A  C_ 
U. B  /\  U. B  C_  A ) )
64, 5sylibr 217 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 375    = wceq 1455    e. wcel 1898   A.wral 2749    C_ wss 3416   U.cuni 4212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ral 2754  df-v 3059  df-in 3423  df-ss 3430  df-uni 4213
This theorem is referenced by:  unimax  4247  shsspwh  26948
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