Theorem aiotavb 44582

Description: The alternate iota over a wff 𝜑 is the universe iff there is no unique value 𝑥 satisfying 𝜑. (Contributed by AV, 25-Aug-2022.)

Assertion

Ref	Expression
aiotavb	⊢ (¬ ∃!𝑥𝜑 ↔ (℩'𝑥𝜑) = V)

Proof of Theorem aiotavb

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		intnex 5262	. . 3 ⊢ (¬ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ∈ V ↔ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = V)
2		df-aiota 44577	. . . . 5 ⊢ (℩'𝑥𝜑) = ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
3	2	eleq1i 2829	. . . 4 ⊢ ((℩'𝑥𝜑) ∈ V ↔ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ∈ V)
4	3	notbii 320	. . 3 ⊢ (¬ (℩'𝑥𝜑) ∈ V ↔ ¬ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ∈ V)
5	2	eqeq1i 2743	. . 3 ⊢ ((℩'𝑥𝜑) = V ↔ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = V)
6	1, 4, 5	3bitr4i 303	. 2 ⊢ (¬ (℩'𝑥𝜑) ∈ V ↔ (℩'𝑥𝜑) = V)
7		aiotaexb 44581	. 2 ⊢ (∃!𝑥𝜑 ↔ (℩'𝑥𝜑) ∈ V)
8	6, 7	xchnxbir 333	1 ⊢ (¬ ∃!𝑥𝜑 ↔ (℩'𝑥𝜑) = V)

Syntax hints:  ¬ wn 3   ↔ wb 205   = wceq 1539   ∈ wcel 2106  ∃!weu 2568  {cab 2715  Vcvv 3432  {csn 4561  ∩ cint 4879  ℩'caiota 44575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-in 3894  df-ss 3904  df-nul 4257  df-sn 4562  df-int 4880  df-aiota 44577

This theorem is referenced by:  dfaiota3  44584  dfafv2  44624