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Theorem unissel 4276
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 457 . 2  |-  ( ( U. A  C_  B  /\  B  e.  A
)  ->  U. A  C_  B )
2 elssuni 4275 . . 3  |-  ( B  e.  A  ->  B  C_ 
U. A )
32adantl 466 . 2  |-  ( ( U. A  C_  B  /\  B  e.  A
)  ->  B  C_  U. A
)
41, 3eqssd 3521 1  |-  ( ( U. A  C_  B  /\  B  e.  A
)  ->  U. A  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    C_ wss 3476   U.cuni 4245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115  df-in 3483  df-ss 3490  df-uni 4246
This theorem is referenced by:  elpwuni  4413  istps2OLD  19189  mretopd  19359  toponmre  19360  neiptopuni  19397  filunibas  20117  unicls  27521  unidmvol  27840
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