Theorem uniintabim 3868

Description: The union and the intersection of a class abstraction are equal if there is a unique satisfying value of ph ( x ). (Contributed by Jim Kingdon, 14-Aug-2018.)

Assertion

Ref	Expression
uniintabim	|- ( E! x ph -> U. { x | ph } = |^| { x | ph } )

Proof of Theorem uniintabim

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		euabsn2 3652	. 2 |- ( E! x ph <-> E. y { x | ph } = { y } )
2		uniintsnr 3867	. 2 |- ( E. y { x | ph } = { y } -> U. { x | ph } = |^| { x | ph } )
3	1, 2	sylbi 120	1 |- ( E! x ph -> U. { x | ph } = |^| { x | ph } )

Syntax hints:   -> wi 4   = wceq 1348   E.wex 1485   E!weu 2019   {cab 2156   {csn 3583   U.cuni 3796   |^|cint 3831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-sn 3589  df-pr 3590  df-uni 3797  df-int 3832

This theorem is referenced by:  iotaint  5173