Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  aiotavb Structured version   Visualization version   GIF version

Theorem aiotavb 45798
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.
StepHypRef Expression
1 intnex 5339 . . 3 {𝑦 ∣ {𝑥𝜑} = {𝑦}} ∈ V ↔ {𝑦 ∣ {𝑥𝜑} = {𝑦}} = V)
2 df-aiota 45793 . . . . 5 (℩'𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
32eleq1i 2825 . . . 4 ((℩'𝑥𝜑) ∈ V ↔ {𝑦 ∣ {𝑥𝜑} = {𝑦}} ∈ V)
43notbii 320 . . 3 (¬ (℩'𝑥𝜑) ∈ V ↔ ¬ {𝑦 ∣ {𝑥𝜑} = {𝑦}} ∈ V)
52eqeq1i 2738 . . 3 ((℩'𝑥𝜑) = V ↔ {𝑦 ∣ {𝑥𝜑} = {𝑦}} = V)
61, 4, 53bitr4i 303 . 2 (¬ (℩'𝑥𝜑) ∈ V ↔ (℩'𝑥𝜑) = V)
7 aiotaexb 45797 . 2 (∃!𝑥𝜑 ↔ (℩'𝑥𝜑) ∈ V)
86, 7xchnxbir 333 1 (¬ ∃!𝑥𝜑 ↔ (℩'𝑥𝜑) = V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205   = wceq 1542  wcel 2107  ∃!weu 2563  {cab 2710  Vcvv 3475  {csn 4629   cint 4951  ℩'caiota 45791
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-in 3956  df-ss 3966  df-nul 4324  df-sn 4630  df-int 4952  df-aiota 45793
This theorem is referenced by:  dfaiota3  45800  dfafv2  45840
  Copyright terms: Public domain W3C validator