Theorem uniintab 4916

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 ↔ ∪ {𝑥 ∣ 𝜑} = ∩ {𝑥 ∣ 𝜑})

Proof of Theorem uniintab

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		euabsn2 4663	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
2		uniintsn 4915	. 2 ⊢ (∪ {𝑥 ∣ 𝜑} = ∩ {𝑥 ∣ 𝜑} ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
3	1, 2	bitr4i 280	1 ⊢ (∃!𝑥𝜑 ↔ ∪ {𝑥 ∣ 𝜑} = ∩ {𝑥 ∣ 𝜑})

Syntax hints:   ↔ wb 208   = wceq 1537  ∃wex 1780  ∃!weu 2653  {cab 2801  {csn 4569  ∪ cuni 4840  ∩ cint 4878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-sn 4570  df-pr 4572  df-uni 4841  df-int 4879

This theorem is referenced by:  iotaint  6333  reuabaiotaiota  43294