Theorem iotaex 5785

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 5779	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) = 𝑧)
2	1	eqcomd 2616	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → 𝑧 = (℩𝑥𝜑))
3	2	eximi 1752	. . 3 ⊢ (∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∃𝑧 𝑧 = (℩𝑥𝜑))
4		df-eu 2462	. . 3 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧))
5		isset 3180	. . 3 ⊢ ((℩𝑥𝜑) ∈ V ↔ ∃𝑧 𝑧 = (℩𝑥𝜑))
6	3, 4, 5	3imtr4i 280	. 2 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
7		iotanul 5783	. . 3 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
8		0ex 4718	. . 3 ⊢ ∅ ∈ V
9	7, 8	syl6eqel 2696	. 2 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
10	6, 9	pm2.61i 175	1 ⊢ (℩𝑥𝜑) ∈ V

Syntax hints:  ¬ wn 3   ↔ wb 195  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃!weu 2458  Vcvv 3173  ∅c0 3874  ℩cio 5766

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126  df-pr 4128  df-uni 4373  df-iota 5768

This theorem is referenced by:  iota4an  5787  fvex  6113  riotaex  6515  erov  7731  iunfictbso  8820  isf32lem9  9066  sumex  14266  prodex  14476  pcval  15387  grpidval  17083  fn0g  17085  gsumvalx  17093  psgnfn  17744  psgnval  17750  dchrptlem1  24789  lgsdchrval  24879  lgsdchr  24880  bnj1366  30154  bj-finsumval0  32324  ellimciota  38681  fourierdlem36  39036