Theorem iotanul 6538

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 2573	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧))
2		dfiota2 6514	. . 3 ⊢ (℩𝑥𝜑) = ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)}
3		alnex 1780	. . . . . 6 ⊢ (∀𝑧 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧))
4		dfnul2 4335	. . . . . . 7 ⊢ ∅ = {𝑧 ∣ ¬ 𝑧 = 𝑧}
5		equid 2010	. . . . . . . . . . . 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 1810	. . . . . . . 8 ⊢ (∀𝑧 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∀𝑧(¬ 𝑧 = 𝑧 ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)))
10		abbi 2806	. . . . . . . 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 2789	. . . . . 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 4919	. . . 4 ⊢ (¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} = ∪ ∅)
15		uni0 4934	. . . 4 ⊢ ∪ ∅ = ∅
16	14, 15	eqtrdi 2792	. . 3 ⊢ (¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} = ∅)
17	2, 16	eqtrid 2788	. 2 ⊢ (¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) = ∅)
18	1, 17	sylnbi 330	1 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206  ∀wal 1537   = wceq 1539  ∃wex 1778  ∃!weu 2567  {cab 2713  ∅c0 4332  ∪ cuni 4906  ℩cio 6511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-v 3481  df-dif 3953  df-ss 3967  df-nul 4333  df-sn 4626  df-uni 4907  df-iota 6513

This theorem is referenced by:  iotassuniOLD  6539  iotaexOLD  6540  iotan0  6550  dfiota4  6552  csbiota  6553  tz6.12-2  6893  dffv3  6901  csbriota  7404  riotaund  7428  isf32lem9  10402  grpidval  18675  0g0  18678  iota0ndef  47056  iotan0aiotaex  47110