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Theorem ssunieq 3708
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 3703 . . 3  |-  ( A  e.  B  ->  A  C_ 
U. B )
2 unissb 3705 . . . 4  |-  ( U. B  C_  A  <->  A. x  e.  B  x  C_  A
)
32biimpri 132 . . 3  |-  ( A. x  e.  B  x  C_  A  ->  U. B  C_  A )
41, 3anim12i 332 . 2  |-  ( ( A  e.  B  /\  A. x  e.  B  x 
C_  A )  -> 
( A  C_  U. B  /\  U. B  C_  A
) )
5 eqss 3054 . 2  |-  ( A  =  U. B  <->  ( A  C_ 
U. B  /\  U. B  C_  A ) )
64, 5sylibr 133 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 103    = wceq 1296    e. wcel 1445   A.wral 2370    C_ wss 3013   U.cuni 3675
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-v 2635  df-in 3019  df-ss 3026  df-uni 3676
This theorem is referenced by:  unimax  3709  hashinfuni  10300  hashennnuni  10302
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