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Theorem iotanul 5854
 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.
StepHypRef Expression
1 df-eu 2472 . 2 (∃!𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
2 dfiota2 5840 . . 3 (℩𝑥𝜑) = {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)}
3 alnex 1704 . . . . . 6 (∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧) ↔ ¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
4 dfnul2 3909 . . . . . . 7 ∅ = {𝑧 ∣ ¬ 𝑧 = 𝑧}
5 equid 1937 . . . . . . . . . . . 12 𝑧 = 𝑧
65tbt 359 . . . . . . . . . . 11 (¬ ∀𝑥(𝜑𝑥 = 𝑧) ↔ (¬ ∀𝑥(𝜑𝑥 = 𝑧) ↔ 𝑧 = 𝑧))
76biimpi 206 . . . . . . . . . 10 (¬ ∀𝑥(𝜑𝑥 = 𝑧) → (¬ ∀𝑥(𝜑𝑥 = 𝑧) ↔ 𝑧 = 𝑧))
87con1bid 345 . . . . . . . . 9 (¬ ∀𝑥(𝜑𝑥 = 𝑧) → (¬ 𝑧 = 𝑧 ↔ ∀𝑥(𝜑𝑥 = 𝑧)))
98alimi 1737 . . . . . . . 8 (∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧) → ∀𝑧𝑧 = 𝑧 ↔ ∀𝑥(𝜑𝑥 = 𝑧)))
10 abbi 2735 . . . . . . . 8 (∀𝑧𝑧 = 𝑧 ↔ ∀𝑥(𝜑𝑥 = 𝑧)) ↔ {𝑧 ∣ ¬ 𝑧 = 𝑧} = {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)})
119, 10sylib 208 . . . . . . 7 (∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ¬ 𝑧 = 𝑧} = {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)})
124, 11syl5req 2667 . . . . . 6 (∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = ∅)
133, 12sylbir 225 . . . . 5 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = ∅)
1413unieqd 4437 . . . 4 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = ∅)
15 uni0 4456 . . . 4 ∅ = ∅
1614, 15syl6eq 2670 . . 3 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = ∅)
172, 16syl5eq 2666 . 2 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → (℩𝑥𝜑) = ∅)
181, 17sylnbi 320 1 (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196  ∀wal 1479   = wceq 1481  ∃wex 1702  ∃!weu 2468  {cab 2606  ∅c0 3907  ∪ cuni 4427  ℩cio 5837 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-v 3197  df-dif 3570  df-in 3574  df-ss 3581  df-nul 3908  df-sn 4169  df-uni 4428  df-iota 5839 This theorem is referenced by:  iotassuni  5855  iotaex  5856  dfiota4  5867  dfiota4OLD  5868  csbiota  5869  tz6.12-2  6169  dffv3  6174  csbriota  6608  riotaund  6632  isf32lem9  9168  grpidval  17241  0g0  17244
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