Theorem uniintabim 3808

Description: The union and the intersection of a class abstraction are equal if there is a unique satisfying value of 𝜑(𝑥). (Contributed by Jim Kingdon, 14-Aug-2018.)

Assertion

Ref	Expression
uniintabim	⊢ (∃!𝑥𝜑 → ∪ {𝑥 ∣ 𝜑} = ∩ {𝑥 ∣ 𝜑})

Proof of Theorem uniintabim

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		euabsn2 3592	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
2		uniintsnr 3807	. 2 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → ∪ {𝑥 ∣ 𝜑} = ∩ {𝑥 ∣ 𝜑})
3	1, 2	sylbi 120	1 ⊢ (∃!𝑥𝜑 → ∪ {𝑥 ∣ 𝜑} = ∩ {𝑥 ∣ 𝜑})

Syntax hints:   → wi 4   = wceq 1331  ∃wex 1468  ∃!weu 1999  {cab 2125  {csn 3527  ∪ cuni 3736  ∩ cint 3771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-sn 3533  df-pr 3534  df-uni 3737  df-int 3772

This theorem is referenced by:  iotaint  5101