Theorem iotaex 6457

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 2144, ax-11 2160, ax-12 2180. (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 6452	. . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = 𝑦)
2		vex 3440	. . . 4 ⊢ 𝑦 ∈ V
3	1, 2	eqeltrdi 2839	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) ∈ V)
4	3	exlimiv 1931	. 2 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) ∈ V)
5		iotanul2 6454	. . 3 ⊢ (¬ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∅)
6		0ex 5245	. . 3 ⊢ ∅ ∈ V
7	5, 6	eqeltrdi 2839	. 2 ⊢ (¬ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) ∈ V)
8	4, 7	pm2.61i 182	1 ⊢ (℩𝑥𝜑) ∈ V

Syntax hints:  ¬ wn 3   = wceq 1541  ∃wex 1780   ∈ wcel 2111  {cab 2709  Vcvv 3436  ∅c0 4283  {csn 4576  ℩cio 6435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-nul 5244

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-v 3438  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-sn 4577  df-pr 4579  df-uni 4860  df-iota 6437

This theorem is referenced by:  iota4an  6463  fvex  6835  riotaex  7307  erov  8738  iunfictbso  10005  isf32lem9  10252  sumex  15595  prodex  15812  pcval  16756  grpidval  18569  fn0g  18571  gsumvalx  18584  psgnfn  19414  psgnval  19420  dchrptlem1  27203  lgsdchrval  27293  lgsdchr  27294  nosupno  27643  nosupdm  27644  nosupbday  27645  nosupfv  27646  nosupres  27647  nosupbnd1lem1  27648  noinfno  27658  noinfdm  27659  noinffv  27661  bnj1366  34839  bj-finsumval0  37325  ellimciota  45660  fourierdlem36  46187  eubrdm  47073  dfatafv2ex  47250  afv2ex  47251  funressndmafv2rn  47260