Theorem uniintab 4938

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 4679	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
2		uniintsn 4937	. 2 ⊢ (∪ {𝑥 ∣ 𝜑} = ∩ {𝑥 ∣ 𝜑} ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
3	1, 2	bitr4i 278	1 ⊢ (∃!𝑥𝜑 ↔ ∪ {𝑥 ∣ 𝜑} = ∩ {𝑥 ∣ 𝜑})

Syntax hints:   ↔ wb 206   = wceq 1541  ∃wex 1780  ∃!weu 2565  {cab 2711  {csn 4577  ∪ cuni 4860  ∩ cint 4899

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-sn 4578  df-pr 4580  df-uni 4861  df-int 4900

This theorem is referenced by:  iotaint  6467  reuabaiotaiota  47249