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Theorem unissel 3872
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 443 . 2  |-  ( ( U. A  C_  B  /\  B  e.  A
)  ->  U. A  C_  B )
2 elssuni 3871 . . 3  |-  ( B  e.  A  ->  B  C_ 
U. A )
32adantl 452 . 2  |-  ( ( U. A  C_  B  /\  B  e.  A
)  ->  B  C_  U. A
)
41, 3eqssd 3209 1  |-  ( ( U. A  C_  B  /\  B  e.  A
)  ->  U. A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    C_ wss 3165   U.cuni 3843
This theorem is referenced by:  elpwuni  4005  istps2OLD  16675  mretopd  16845  toponmre  16846  filunibas  17592  unidmvol  23210
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-in 3172  df-ss 3179  df-uni 3844
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