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Theorem unissel 3986
Description: Condition turning a subclass relationship for union into an equality. (Contributed by NM, 18-Jul-2006.)
Assertion
Ref Expression
unissel  |-  ( ( U. A  C_  B  /\  B  e.  A
)  ->  U. A  =  B )

Proof of Theorem unissel
StepHypRef Expression
1 simpl 444 . 2  |-  ( ( U. A  C_  B  /\  B  e.  A
)  ->  U. A  C_  B )
2 elssuni 3985 . . 3  |-  ( B  e.  A  ->  B  C_ 
U. A )
32adantl 453 . 2  |-  ( ( U. A  C_  B  /\  B  e.  A
)  ->  B  C_  U. A
)
41, 3eqssd 3308 1  |-  ( ( U. A  C_  B  /\  B  e.  A
)  ->  U. A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    C_ wss 3263   U.cuni 3957
This theorem is referenced by:  elpwuni  4119  istps2OLD  16909  mretopd  17079  toponmre  17080  neiptopuni  17117  filunibas  17834  unicls  24105  unidmvol  24378
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 2368
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 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-v 2901  df-in 3270  df-ss 3277  df-uni 3958
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