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Theorem uniintab 4080
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 3867 . 2  |-  ( E! x ph  <->  E. y { x  |  ph }  =  { y } )
2 uniintsn 4079 . 2  |-  ( U. { x  |  ph }  =  |^| { x  | 
ph }  <->  E. y { x  |  ph }  =  { y } )
31, 2bitr4i 244 1  |-  ( E! x ph  <->  U. { x  |  ph }  =  |^| { x  |  ph }
)
Colors of variables: wff set class
Syntax hints:    <-> wb 177   E.wex 1550    = wceq 1652   E!weu 2280   {cab 2421   {csn 3806   U.cuni 4007   |^|cint 4042
This theorem is referenced by:  iotaint  5423
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-sn 3812  df-pr 3813  df-uni 4008  df-int 4043
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