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Theorem uniintab 4991
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 4730 . 2 (∃!𝑥𝜑 ↔ ∃𝑦{𝑥𝜑} = {𝑦})
2 uniintsn 4990 . 2 ( {𝑥𝜑} = {𝑥𝜑} ↔ ∃𝑦{𝑥𝜑} = {𝑦})
31, 2bitr4i 278 1 (∃!𝑥𝜑 {𝑥𝜑} = {𝑥𝜑})
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1537  wex 1776  ∃!weu 2566  {cab 2712  {csn 4631   cuni 4912   cint 4951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-sn 4632  df-pr 4634  df-uni 4913  df-int 4952
This theorem is referenced by:  iotaint  6539  reuabaiotaiota  47037
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