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Theorem uniintab 3916
Description: The union and the intersection of a class abstraction are equal exactly when there is a unique satisfying value of  ph ( x ). (Contributed by Mario Carneiro, 24-Dec-2016.)
Assertion
Ref Expression
uniintab  |-  ( E! x ph  <->  U. { x  |  ph }  =  |^| { x  |  ph }
)

Proof of Theorem uniintab
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 euabsn2 3711 . 2  |-  ( E! x ph  <->  E. y { x  |  ph }  =  { y } )
2 uniintsn 3915 . 2  |-  ( U. { x  |  ph }  =  |^| { x  | 
ph }  <->  E. y { x  |  ph }  =  { y } )
31, 2bitr4i 243 1  |-  ( E! x ph  <->  U. { x  |  ph }  =  |^| { x  |  ph }
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176   E.wex 1531    = wceq 1632   E!weu 2156   {cab 2282   {csn 3653   U.cuni 3843   |^|cint 3878
This theorem is referenced by:  iotaint  5248
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-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-sn 3659  df-pr 3660  df-uni 3844  df-int 3879
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