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Theorem uniintab 4985
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 4724 . 2 (∃!𝑥𝜑 ↔ ∃𝑦{𝑥𝜑} = {𝑦})
2 uniintsn 4984 . 2 ( {𝑥𝜑} = {𝑥𝜑} ↔ ∃𝑦{𝑥𝜑} = {𝑦})
31, 2bitr4i 278 1 (∃!𝑥𝜑 {𝑥𝜑} = {𝑥𝜑})
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1539  wex 1778  ∃!weu 2567  {cab 2713  {csn 4625   cuni 4906   cint 4945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-sn 4626  df-pr 4628  df-uni 4907  df-int 4946
This theorem is referenced by:  iotaint  6536  reuabaiotaiota  47104
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