Theorem uniintabim 3883

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 3663	. 2 |- ( E! x ph <-> E. y { x | ph } = { y } )
2		uniintsnr 3882	. 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 1353   E.wex 1492   E!weu 2026   {cab 2163   {csn 3594   U.cuni 3811   |^|cint 3846

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-sn 3600  df-pr 3601  df-uni 3812  df-int 3847

This theorem is referenced by:  iotaint  5193