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Theorem iotanul 6009
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 2622 . 2 (∃!𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
2 dfiota2 5995 . . 3 (℩𝑥𝜑) = {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)}
3 alnex 1854 . . . . . 6 (∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧) ↔ ¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
4 dfnul2 4065 . . . . . . 7 ∅ = {𝑧 ∣ ¬ 𝑧 = 𝑧}
5 equid 2097 . . . . . . . . . . . 12 𝑧 = 𝑧
65tbt 358 . . . . . . . . . . 11 (¬ ∀𝑥(𝜑𝑥 = 𝑧) ↔ (¬ ∀𝑥(𝜑𝑥 = 𝑧) ↔ 𝑧 = 𝑧))
76biimpi 206 . . . . . . . . . 10 (¬ ∀𝑥(𝜑𝑥 = 𝑧) → (¬ ∀𝑥(𝜑𝑥 = 𝑧) ↔ 𝑧 = 𝑧))
87con1bid 344 . . . . . . . . 9 (¬ ∀𝑥(𝜑𝑥 = 𝑧) → (¬ 𝑧 = 𝑧 ↔ ∀𝑥(𝜑𝑥 = 𝑧)))
98alimi 1887 . . . . . . . 8 (∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧) → ∀𝑧𝑧 = 𝑧 ↔ ∀𝑥(𝜑𝑥 = 𝑧)))
10 abbi 2886 . . . . . . . 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 2818 . . . . . 6 (∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = ∅)
133, 12sylbir 225 . . . . 5 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = ∅)
1413unieqd 4584 . . . 4 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = ∅)
15 uni0 4601 . . . 4 ∅ = ∅
1614, 15syl6eq 2821 . . 3 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = ∅)
172, 16syl5eq 2817 . 2 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → (℩𝑥𝜑) = ∅)
181, 17sylnbi 319 1 (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wal 1629   = wceq 1631  wex 1852  ∃!weu 2618  {cab 2757  c0 4063   cuni 4574  cio 5992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-v 3353  df-dif 3726  df-in 3730  df-ss 3737  df-nul 4064  df-sn 4317  df-uni 4575  df-iota 5994
This theorem is referenced by:  iotassuni  6010  iotaex  6011  dfiota4  6022  dfiota4OLD  6023  csbiota  6024  tz6.12-2  6323  dffv3  6328  csbriota  6766  riotaund  6790  isf32lem9  9385  grpidval  17468  0g0  17471
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