Theorem uniintabim 3911

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 3691	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
2		uniintsnr 3910	. 2 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → ∪ {𝑥 ∣ 𝜑} = ∩ {𝑥 ∣ 𝜑})
3	1, 2	sylbi 121	1 ⊢ (∃!𝑥𝜑 → ∪ {𝑥 ∣ 𝜑} = ∩ {𝑥 ∣ 𝜑})

Syntax hints:   → wi 4   = wceq 1364  ∃wex 1506  ∃!weu 2045  {cab 2182  {csn 3622  ∪ cuni 3839  ∩ cint 3874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-sn 3628  df-pr 3629  df-uni 3840  df-int 3875

This theorem is referenced by:  iotaint  5232