MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uniintab Structured version   Unicode version

Theorem uniintab 4320
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 4098 . 2  |-  ( E! x ph  <->  E. y { x  |  ph }  =  { y } )
2 uniintsn 4319 . 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 1379   E.wex 1596   E!weu 2275   {cab 2452   {csn 4027   U.cuni 4245   |^|cint 4282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-sn 4028  df-pr 4030  df-uni 4246  df-int 4283
This theorem is referenced by:  iotaint  5562
  Copyright terms: Public domain W3C validator