Theorem aiotavb 47645

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 5298	. . 3 ⊢ (¬ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ∈ V ↔ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = V)
2		df-aiota 47640	. . . . 5 ⊢ (℩'𝑥𝜑) = ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
3	2	eleq1i 2852	. . . 4 ⊢ ((℩'𝑥𝜑) ∈ V ↔ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ∈ V)
4	3	notbii 322	. . 3 ⊢ (¬ (℩'𝑥𝜑) ∈ V ↔ ¬ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ∈ V)
5	2	eqeq1i 2766	. . 3 ⊢ ((℩'𝑥𝜑) = V ↔ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = V)
6	1, 4, 5	3bitr4i 305	. 2 ⊢ (¬ (℩'𝑥𝜑) ∈ V ↔ (℩'𝑥𝜑) = V)
7		aiotaexb 47644	. 2 ⊢ (∃!𝑥𝜑 ↔ (℩'𝑥𝜑) ∈ V)
8	6, 7	xchnxbir 335	1 ⊢ (¬ ∃!𝑥𝜑 ↔ (℩'𝑥𝜑) = V)

Syntax hints:  ¬ wn 3   ↔ wb 208   = wceq 1559   ∈ wcel 2141  ∃!weu 2594  {cab 2739  Vcvv 3453  {csn 4579  ∩ cint 4902  ℩'caiota 47638

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-in 3909  df-ss 3919  df-nul 4284  df-sn 4580  df-int 4903  df-aiota 47640

This theorem is referenced by:  dfaiota3  47647  dfafv2  47687