Theorem iotanul 6472

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 2578	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧))
2		dfiota2 6449	. . 3 ⊢ (℩𝑥𝜑) = ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)}
3		alnex 1788	. . . . . 6 ⊢ (∀𝑧 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧))
4		dfnul2 4271	. . . . . . 7 ⊢ ∅ = {𝑧 ∣ ¬ 𝑧 = 𝑧}
5		equid 2019	. . . . . . . . . . . 12 ⊢ 𝑧 = 𝑧
6	5	tbt 370	. . . . . . . . . . 11 ⊢ (¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ (¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ 𝑧 = 𝑧))
7	6	biimpi 217	. . . . . . . . . 10 ⊢ (¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ 𝑧 = 𝑧))
8	7	con1bid 356	. . . . . . . . 9 ⊢ (¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (¬ 𝑧 = 𝑧 ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)))
9	8	alimi 1818	. . . . . . . 8 ⊢ (∀𝑧 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∀𝑧(¬ 𝑧 = 𝑧 ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)))
10		abbi 2805	. . . . . . . 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 2788	. . . . . 6 ⊢ (∀𝑧 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} = ∅)
13	3, 12	sylbir 236	. . . . 5 ⊢ (¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} = ∅)
14	13	unieqd 4858	. . . 4 ⊢ (¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} = ∪ ∅)
15		uni0 4873	. . . 4 ⊢ ∪ ∅ = ∅
16	14, 15	eqtrdi 2791	. . 3 ⊢ (¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} = ∅)
17	2, 16	eqtrid 2787	. 2 ⊢ (¬ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) = ∅)
18	1, 17	sylnbi 331	1 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207  ∀wal 1545   = wceq 1547  ∃wex 1786  ∃!weu 2572  {cab 2718  ∅c0 4268  ∪ cuni 4845  ℩cio 6446

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-v 3434  df-dif 3893  df-ss 3907  df-nul 4269  df-sn 4563  df-uni 4846  df-iota 6448

This theorem is referenced by:  iotan0  6482  dfiota4  6484  csbiota  6485  tz6.12-2OLD  6822  dffv3  6830  csbriota  7335  riotaund  7359  isf32lem9  10281  grpidval  18627  0g0  18630  iota0ndef  47509  iotan0aiotaex  47563