Theorem iotaex 6412

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.)

Assertion

Ref	Expression
iotaex	⊢ (℩𝑥𝜑) ∈ V

Proof of Theorem iotaex

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iotaval 6406	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) = 𝑧)
2	1	eqcomd 2746	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → 𝑧 = (℩𝑥𝜑))
3	2	eximi 1841	. . 3 ⊢ (∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∃𝑧 𝑧 = (℩𝑥𝜑))
4		eu6 2576	. . 3 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧))
5		isset 3444	. . 3 ⊢ ((℩𝑥𝜑) ∈ V ↔ ∃𝑧 𝑧 = (℩𝑥𝜑))
6	3, 4, 5	3imtr4i 292	. 2 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
7		iotanul 6410	. . 3 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
8		0ex 5235	. . 3 ⊢ ∅ ∈ V
9	7, 8	eqeltrdi 2849	. 2 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
10	6, 9	pm2.61i 182	1 ⊢ (℩𝑥𝜑) ∈ V

Syntax hints:  ¬ wn 3   ↔ wb 205  ∀wal 1540   = wceq 1542  ∃wex 1786   ∈ wcel 2110  ∃!weu 2570  Vcvv 3431  ∅c0 4262  ℩cio 6388

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-nul 5234

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-ral 3071  df-rex 3072  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-sn 4568  df-pr 4570  df-uni 4846  df-iota 6390

This theorem is referenced by:  iota4an  6414  fvex  6784  riotaex  7232  erov  8586  iunfictbso  9871  isf32lem9  10118  sumex  15397  prodex  15615  pcval  16543  grpidval  18343  fn0g  18345  gsumvalx  18358  psgnfn  19107  psgnval  19113  dchrptlem1  26410  lgsdchrval  26500  lgsdchr  26501  bnj1366  32805  nosupno  33902  nosupdm  33903  nosupbday  33904  nosupfv  33905  nosupres  33906  nosupbnd1lem1  33907  noinfno  33917  noinfdm  33918  noinffv  33920  bj-finsumval0  35452  ellimciota  43126  fourierdlem36  43655  eubrdm  44498  dfatafv2ex  44673  afv2ex  44674  funressndmafv2rn  44683