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Theorem unissel 4220
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 455 . 2  |-  ( ( U. A  C_  B  /\  B  e.  A
)  ->  U. A  C_  B )
2 elssuni 4219 . . 3  |-  ( B  e.  A  ->  B  C_ 
U. A )
32adantl 464 . 2  |-  ( ( U. A  C_  B  /\  B  e.  A
)  ->  B  C_  U. A
)
41, 3eqssd 3458 1  |-  ( ( U. A  C_  B  /\  B  e.  A
)  ->  U. A  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842    C_ wss 3413   U.cuni 4190
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-v 3060  df-in 3420  df-ss 3427  df-uni 4191
This theorem is referenced by:  elpwuni  4361  istps2OLD  19712  mretopd  19884  toponmre  19885  neiptopuni  19922  filunibas  20672  unicls  28324  unidmvol  28663  carsguni  28742
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