Theorem uniintabim 3928

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 3707	. 2 |- ( E! x ph <-> E. y { x | ph } = { y } )
2		uniintsnr 3927	. 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 1373   E.wex 1516   E!weu 2055   {cab 2192   {csn 3638   U.cuni 3856   |^|cint 3891

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3174  df-in 3176  df-sn 3644  df-pr 3645  df-uni 3857  df-int 3892

This theorem is referenced by:  iotaint  5254