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Theorem uniintab 4265
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 4042 . 2  |-  ( E! x ph  <->  E. y { x  |  ph }  =  { y } )
2 uniintsn 4264 . 2  |-  ( U. { x  |  ph }  =  |^| { x  | 
ph }  <->  E. y { x  |  ph }  =  { y } )
31, 2bitr4i 252 1  |-  ( E! x ph  <->  U. { x  |  ph }  =  |^| { x  |  ph }
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1405   E.wex 1633   E!weu 2238   {cab 2387   {csn 3971   U.cuni 4190   |^|cint 4226
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-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-sn 3972  df-pr 3974  df-uni 4191  df-int 4227
This theorem is referenced by:  iotaint  5545
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