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Theorem ssunieq 3686
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 3681 . . 3  |-  ( A  e.  B  ->  A  C_ 
U. B )
2 unissb 3683 . . . 4  |-  ( U. B  C_  A  <->  A. x  e.  B  x  C_  A
)
32biimpri 131 . . 3  |-  ( A. x  e.  B  x  C_  A  ->  U. B  C_  A )
41, 3anim12i 331 . 2  |-  ( ( A  e.  B  /\  A. x  e.  B  x 
C_  A )  -> 
( A  C_  U. B  /\  U. B  C_  A
) )
5 eqss 3040 . 2  |-  ( A  =  U. B  <->  ( A  C_ 
U. B  /\  U. B  C_  A ) )
64, 5sylibr 132 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 102    = wceq 1289    e. wcel 1438   A.wral 2359    C_ wss 2999   U.cuni 3653
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-in 3005  df-ss 3012  df-uni 3654
This theorem is referenced by:  unimax  3687  hashinfuni  10185  hashennnuni  10187
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