Theorem uniintabim 3860

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 3644	. 2 |- ( E! x ph <-> E. y { x | ph } = { y } )
2		uniintsnr 3859	. 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 1343   E.wex 1480   E!weu 2014   {cab 2151   {csn 3575   U.cuni 3788   |^|cint 3823

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ral 2448  df-rex 2449  df-v 2727  df-un 3119  df-in 3121  df-sn 3581  df-pr 3582  df-uni 3789  df-int 3824

This theorem is referenced by:  iotaint  5165