Theorem iotaex 6501

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 2178, ax-11 2194, ax-12 2215. (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 6496	. . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = 𝑦)
2		vex 3461	. . . 4 ⊢ 𝑦 ∈ V
3	1, 2	eqeltrdi 2873	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) ∈ V)
4	3	exlimiv 1953	. 2 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) ∈ V)
5		iotanul2 6498	. . 3 ⊢ (¬ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∅)
6		0ex 5261	. . 3 ⊢ ∅ ∈ V
7	5, 6	eqeltrdi 2873	. 2 ⊢ (¬ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) ∈ V)
8	4, 7	pm2.61i 184	1 ⊢ (℩𝑥𝜑) ∈ V

Syntax hints:  ¬ wn 3   = wceq 1563  ∃wex 1802   ∈ wcel 2145  {cab 2743  Vcvv 3457  ∅c0 4288  {csn 4585  ℩cio 6479

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-nul 5260

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-v 3459  df-dif 3910  df-un 3912  df-ss 3924  df-nul 4289  df-sn 4586  df-pr 4588  df-uni 4868  df-iota 6481

This theorem is referenced by:  iota4an  6507  fvex  6884  riotaex  7361  erov  8800  iunfictbso  10086  isf32lem9  10333  sumex  15727  prodex  15947  pcval  16892  grpidval  18707  fn0g  18709  gsumvalx  18722  psgnfn  19559  psgnval  19565  dchrptlem1  27382  lgsdchrval  27472  lgsdchr  27473  nosupno  27821  nosupdm  27822  nosupbday  27823  nosupfv  27824  nosupres  27825  nosupbnd1lem1  27826  noinfno  27836  noinfdm  27837  noinffv  27839  bnj1366  35129  bj-finsumval0  37784  preex  38998  ellimciota  46189  fourierdlem36  46716  eubrdm  47629  dfatafv2ex  47806  afv2ex  47807  funressndmafv2rn  47816