Theorem iotaex 6514

Description: Theorem 8.23 in [Quine] p. 58. This theorem proves the existence of the ℩ class under our definition. (Contributed by Andrew Salmon, 11-Jul-2011.) Remove dependency on ax-10 2176, ax-11 2192, ax-12 2213. (Revised by SN, 6-Nov-2024.)

Assertion

Ref	Expression
iotaex	⊢ (℩𝑥𝜑) ∈ V

Proof of Theorem iotaex

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iotaval2 6509	. . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = 𝑦)
2		vex 3459	. . . 4 ⊢ 𝑦 ∈ V
3	1, 2	eqeltrdi 2871	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) ∈ V)
4	3	exlimiv 1960	. 2 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) ∈ V)
5		iotanul2 6511	. . 3 ⊢ (¬ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∅)
6		0ex 5271	. . 3 ⊢ ∅ ∈ V
7	5, 6	eqeltrdi 2871	. 2 ⊢ (¬ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) ∈ V)
8	4, 7	pm2.61i 184	1 ⊢ (℩𝑥𝜑) ∈ V

Syntax hints:  ¬ wn 3   = wceq 1570  ∃wex 1809   ∈ wcel 2143  {cab 2741  Vcvv 3455  ∅c0 4287  {csn 4590  ℩cio 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-nul 5270

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-v 3457  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-sn 4591  df-pr 4593  df-uni 4874  df-iota 6494

This theorem is referenced by:  iota4an  6520  fvex  6896  riotaex  7373  erov  8813  iunfictbso  10099  isf32lem9  10346  sumex  15741  prodex  15961  pcval  16905  grpidval  18720  fn0g  18722  gsumvalx  18735  psgnfn  19572  psgnval  19578  dchrptlem1  27409  lgsdchrval  27499  lgsdchr  27500  nosupno  27848  nosupdm  27849  nosupbday  27850  nosupfv  27851  nosupres  27852  nosupbnd1lem1  27853  noinfno  27863  noinfdm  27864  noinffv  27866  bnj1366  35198  bj-finsumval0  37910  preex  39122  ellimciota  46313  fourierdlem36  46840  eubrdm  47756  dfatafv2ex  47933  afv2ex  47934  funressndmafv2rn  47943