Theorem uniintab 4515

Description: The union and the intersection of a class abstraction are equal exactly when there is a unique satisfying value of ph ( x ). (Contributed by Mario Carneiro, 24-Dec-2016.)

Assertion

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

Proof of Theorem uniintab

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		euabsn2 4260	. 2 |- ( E! x ph <-> E. y { x | ph } = { y } )
2		uniintsn 4514	. 2 |- ( U. { x | ph } = |^| { x | ph } <-> E. y { x | ph } = { y } )
3	1, 2	bitr4i 267	1 |- ( E! x ph <-> U. { x | ph } = |^| { x | ph } )

Syntax hints:   <-> wb 196   = wceq 1483   E.wex 1704   E!weu 2470   {cab 2608   {csn 4177   U.cuni 4436   |^|cint 4475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-pr 4180  df-uni 4437  df-int 4476

This theorem is referenced by:  iotaint  5864