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Theorem uniintab 3964
 Description: The union and the intersection of a class abstraction are equal exactly when there is a unique satisfying value of φ(x). (Contributed by Mario Carneiro, 24-Dec-2016.)
Assertion
Ref Expression
uniintab (∃!xφ{x φ} = {x φ})

Proof of Theorem uniintab
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 euabsn2 3791 . 2 (∃!xφy{x φ} = {y})
2 uniintsn 3963 . 2 ({x φ} = {x φ} ↔ y{x φ} = {y})
31, 2bitr4i 243 1 (∃!xφ{x φ} = {x φ})
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176  ∃wex 1541   = wceq 1642  ∃!weu 2204  {cab 2339  {csn 3737  ∪cuni 3891  ∩cint 3926 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927 This theorem is referenced by:  iotaint  4352
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