Theorem iotaex 6536

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 2180, ax-11 2197, ax-12 2216. (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 6530	. . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = 𝑦)
2		vex 3458	. . . 4 ⊢ 𝑦 ∈ V
3	1, 2	eqeltrdi 2872	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) ∈ V)
4	3	exlimiv 1965	. 2 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) ∈ V)
5		iotanul2 6532	. . 3 ⊢ (¬ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∅)
6		0ex 5265	. . 3 ⊢ ∅ ∈ V
7	5, 6	eqeltrdi 2872	. 2 ⊢ (¬ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) ∈ V)
8	4, 7	pm2.61i 184	1 ⊢ (℩𝑥𝜑) ∈ V

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741  Vcvv 3454  ∅c0 4281  {csn 4588  ℩cio 6513

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-nul 5264

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-sn 4589  df-pr 4591  df-uni 4871  df-iota 6515

This theorem is used by:  iota4an  6542  iotan0  6550  fvex  6924  riotaex  7407  erov  8866  iunfictbso  10223  isf32lem9  10469  sumex  15887  prodex  16106  pcval  17054  grpidval  18872  fn0g  18875  gsumvalx  18897  psgnfn  19748  psgnval  19754  dchrptlem1  27624  lgsdchrval  27714  lgsdchr  27715  nosupno  28093  nosupdm  28094  nosupbday  28095  nosupfv  28096  nosupres  28097  nosupbnd1lem1  28098  noinfno  28108  noinfdm  28109  noinffv  28111  bnj1366  35542  bj-finsumval0  38411  preex  39627  ellimciota  46841  fourierdlem36  47368  eubrdm  48321  dfatafv2ex  48498  afv2ex  48499  funressndmafv2rn  48508