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Theorem uniintab 4450
 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 4204 . 2 (∃!𝑥𝜑 ↔ ∃𝑦{𝑥𝜑} = {𝑦})
2 uniintsn 4449 . 2 ( {𝑥𝜑} = {𝑥𝜑} ↔ ∃𝑦{𝑥𝜑} = {𝑦})
31, 2bitr4i 266 1 (∃!𝑥𝜑 {𝑥𝜑} = {𝑥𝜑})
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475  ∃wex 1695  ∃!weu 2458  {cab 2596  {csn 4125  ∪ cuni 4372  ∩ cint 4410 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126  df-pr 4128  df-uni 4373  df-int 4411 This theorem is referenced by:  iotaint  5781
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