Theorem aiotavb 47560

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 5280	. . 3 ⊢ (¬ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ∈ V ↔ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = V)
2		df-aiota 47555	. . . . 5 ⊢ (℩'𝑥𝜑) = ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
3	2	eleq1i 2831	. . . 4 ⊢ ((℩'𝑥𝜑) ∈ V ↔ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ∈ V)
4	3	notbii 321	. . 3 ⊢ (¬ (℩'𝑥𝜑) ∈ V ↔ ¬ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ∈ V)
5	2	eqeq1i 2745	. . 3 ⊢ ((℩'𝑥𝜑) = V ↔ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = V)
6	1, 4, 5	3bitr4i 304	. 2 ⊢ (¬ (℩'𝑥𝜑) ∈ V ↔ (℩'𝑥𝜑) = V)
7		aiotaexb 47559	. 2 ⊢ (∃!𝑥𝜑 ↔ (℩'𝑥𝜑) ∈ V)
8	6, 7	xchnxbir 334	1 ⊢ (¬ ∃!𝑥𝜑 ↔ (℩'𝑥𝜑) = V)

Syntax hints:  ¬ wn 3   ↔ wb 207   = wceq 1547   ∈ wcel 2119  ∃!weu 2572  {cab 2718  Vcvv 3432  {csn 4562  ∩ cint 4884  ℩'caiota 47553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-in 3897  df-ss 3907  df-nul 4269  df-sn 4563  df-int 4885  df-aiota 47555

This theorem is referenced by:  dfaiota3  47562  dfafv2  47602