Theorem iotanul 6514

Description: Theorem 8.22 in [Quine] p. 57. This theorem is the result if there isn't exactly one 𝑥 that satisfies 𝜑. (Contributed by Andrew Salmon, 11-Jul-2011.)

Assertion

Ref	Expression
iotanul	⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)

Proof of Theorem iotanul

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eu6 2574	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧))
2		dfiota2 6490	. . 3 ⊢ (℩𝑥𝜑) = ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)}
3		alnex 1781	. . . . . 6 ⊢ (∀𝑧 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧))
4		dfnul2 4316	. . . . . . 7 ⊢ ∅ = {𝑧 ∣ ¬ 𝑧 = 𝑧}
5		equid 2012	. . . . . . . . . . . 12 ⊢ 𝑧 = 𝑧
6	5	tbt 369	. . . . . . . . . . 11 ⊢ (¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ (¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ 𝑧 = 𝑧))
7	6	biimpi 216	. . . . . . . . . 10 ⊢ (¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ 𝑧 = 𝑧))
8	7	con1bid 355	. . . . . . . . 9 ⊢ (¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (¬ 𝑧 = 𝑧 ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)))
9	8	alimi 1811	. . . . . . . 8 ⊢ (∀𝑧 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∀𝑧(¬ 𝑧 = 𝑧 ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)))
10		abbi 2801	. . . . . . . 8 ⊢ (∀𝑧(¬ 𝑧 = 𝑧 ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)) → {𝑧 ∣ ¬ 𝑧 = 𝑧} = {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)})
11	9, 10	syl 17	. . . . . . 7 ⊢ (∀𝑧 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → {𝑧 ∣ ¬ 𝑧 = 𝑧} = {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)})
12	4, 11	eqtr2id 2784	. . . . . 6 ⊢ (∀𝑧 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} = ∅)
13	3, 12	sylbir 235	. . . . 5 ⊢ (¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} = ∅)
14	13	unieqd 4901	. . . 4 ⊢ (¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} = ∪ ∅)
15		uni0 4916	. . . 4 ⊢ ∪ ∅ = ∅
16	14, 15	eqtrdi 2787	. . 3 ⊢ (¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} = ∅)
17	2, 16	eqtrid 2783	. 2 ⊢ (¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) = ∅)
18	1, 17	sylnbi 330	1 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206  ∀wal 1538   = wceq 1540  ∃wex 1779  ∃!weu 2568  {cab 2714  ∅c0 4313  ∪ cuni 4888  ℩cio 6487

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-v 3466  df-dif 3934  df-ss 3948  df-nul 4314  df-sn 4607  df-uni 4889  df-iota 6489

This theorem is referenced by:  iotassuniOLD  6515  iotaexOLD  6516  iotan0  6526  dfiota4  6528  csbiota  6529  tz6.12-2  6869  dffv3  6877  csbriota  7382  riotaund  7406  isf32lem9  10380  grpidval  18644  0g0  18647  iota0ndef  47035  iotan0aiotaex  47089