Theorem iotaex 6398

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 6392	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) = 𝑧)
2	1	eqcomd 2744	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → 𝑧 = (℩𝑥𝜑))
3	2	eximi 1838	. . 3 ⊢ (∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∃𝑧 𝑧 = (℩𝑥𝜑))
4		eu6 2574	. . 3 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧))
5		isset 3435	. . 3 ⊢ ((℩𝑥𝜑) ∈ V ↔ ∃𝑧 𝑧 = (℩𝑥𝜑))
6	3, 4, 5	3imtr4i 291	. 2 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
7		iotanul 6396	. . 3 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
8		0ex 5226	. . 3 ⊢ ∅ ∈ V
9	7, 8	eqeltrdi 2847	. 2 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
10	6, 9	pm2.61i 182	1 ⊢ (℩𝑥𝜑) ∈ V

Syntax hints:  ¬ wn 3   ↔ wb 205  ∀wal 1537   = wceq 1539  ∃wex 1783   ∈ wcel 2108  ∃!weu 2568  Vcvv 3422  ∅c0 4253  ℩cio 6374

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-nul 5225

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-sn 4559  df-pr 4561  df-uni 4837  df-iota 6376

This theorem is referenced by:  iota4an  6400  fvex  6769  riotaex  7216  erov  8561  iunfictbso  9801  isf32lem9  10048  sumex  15327  prodex  15545  pcval  16473  grpidval  18260  fn0g  18262  gsumvalx  18275  psgnfn  19024  psgnval  19030  dchrptlem1  26317  lgsdchrval  26407  lgsdchr  26408  bnj1366  32709  nosupno  33833  nosupdm  33834  nosupbday  33835  nosupfv  33836  nosupres  33837  nosupbnd1lem1  33838  noinfno  33848  noinfdm  33849  noinffv  33851  bj-finsumval0  35383  ellimciota  43045  fourierdlem36  43574  eubrdm  44417  dfatafv2ex  44592  afv2ex  44593  funressndmafv2rn  44602