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Theorem uniintab 3900
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 3698 . 2  |-  ( E! x ph  <->  E. y { x  |  ph }  =  { y } )
2 uniintsn 3899 . 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 1528    = wceq 1623   E!weu 2143   {cab 2269   {csn 3640   U.cuni 3827   |^|cint 3862
This theorem is referenced by:  iotaint  5232
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-sn 3646  df-pr 3647  df-uni 3828  df-int 3863
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