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Theorem iotanul 6317
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 eu6 2575 . 2 (∃!𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
2 dfiota2 6298 . . 3 (℩𝑥𝜑) = {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)}
3 alnex 1788 . . . . . 6 (∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧) ↔ ¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
4 dfnul2 4214 . . . . . . 7 ∅ = {𝑧 ∣ ¬ 𝑧 = 𝑧}
5 equid 2024 . . . . . . . . . . . 12 𝑧 = 𝑧
65tbt 373 . . . . . . . . . . 11 (¬ ∀𝑥(𝜑𝑥 = 𝑧) ↔ (¬ ∀𝑥(𝜑𝑥 = 𝑧) ↔ 𝑧 = 𝑧))
76biimpi 219 . . . . . . . . . 10 (¬ ∀𝑥(𝜑𝑥 = 𝑧) → (¬ ∀𝑥(𝜑𝑥 = 𝑧) ↔ 𝑧 = 𝑧))
87con1bid 359 . . . . . . . . 9 (¬ ∀𝑥(𝜑𝑥 = 𝑧) → (¬ 𝑧 = 𝑧 ↔ ∀𝑥(𝜑𝑥 = 𝑧)))
98alimi 1818 . . . . . . . 8 (∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧) → ∀𝑧𝑧 = 𝑧 ↔ ∀𝑥(𝜑𝑥 = 𝑧)))
10 abbi1 2801 . . . . . . . 8 (∀𝑧𝑧 = 𝑧 ↔ ∀𝑥(𝜑𝑥 = 𝑧)) → {𝑧 ∣ ¬ 𝑧 = 𝑧} = {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)})
119, 10syl 17 . . . . . . 7 (∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ¬ 𝑧 = 𝑧} = {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)})
124, 11eqtr2id 2786 . . . . . 6 (∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = ∅)
133, 12sylbir 238 . . . . 5 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = ∅)
1413unieqd 4810 . . . 4 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = ∅)
15 uni0 4826 . . . 4 ∅ = ∅
1614, 15eqtrdi 2789 . . 3 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = ∅)
172, 16syl5eq 2785 . 2 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → (℩𝑥𝜑) = ∅)
181, 17sylnbi 333 1 (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wal 1540   = wceq 1542  wex 1786  ∃!weu 2569  {cab 2716  c0 4211   cuni 4796  cio 6295
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 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-v 3400  df-dif 3846  df-in 3850  df-ss 3860  df-nul 4212  df-sn 4517  df-uni 4797  df-iota 6297
This theorem is referenced by:  iotassuni  6318  iotaex  6319  iotan0  6329  dfiota4  6331  csbiota  6332  tz6.12-2  6663  dffv3  6670  csbriota  7143  riotaund  7167  isf32lem9  9861  grpidval  17987  0g0  17990  iota0ndef  44072  iotan0aiotaex  44117
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