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Theorem uniintab 4887
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 {𝑥𝜑} = {𝑥𝜑})

Proof of Theorem uniintab
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 euabsn2 4634 . 2 (∃!𝑥𝜑 ↔ ∃𝑦{𝑥𝜑} = {𝑦})
2 uniintsn 4886 . 2 ( {𝑥𝜑} = {𝑥𝜑} ↔ ∃𝑦{𝑥𝜑} = {𝑦})
31, 2bitr4i 281 1 (∃!𝑥𝜑 {𝑥𝜑} = {𝑥𝜑})
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1538  wex 1781  ∃!weu 2653  {cab 2799  {csn 4540   cuni 4811   cint 4849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-sn 4541  df-pr 4543  df-uni 4812  df-int 4850
This theorem is referenced by:  iotaint  6304  reuabaiotaiota  43437
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