Theorem uniintabim 3921

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 3701	. 2 |- ( E! x ph <-> E. y { x | ph } = { y } )
2		uniintsnr 3920	. 2 |- ( E. y { x | ph } = { y } -> U. { x | ph } = |^| { x | ph } )
3	1, 2	sylbi 121	1 |- ( E! x ph -> U. { x | ph } = |^| { x | ph } )

Syntax hints:   -> wi 4   = wceq 1372   E.wex 1514   E!weu 2053   {cab 2190   {csn 3632   U.cuni 3849   |^|cint 3884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-sn 3638  df-pr 3639  df-uni 3850  df-int 3885

This theorem is referenced by:  iotaint  5244