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Theorem iotanul 5783
 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 2462 . 2 (∃!𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
2 dfiota2 5769 . . 3 (℩𝑥𝜑) = {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)}
3 alnex 1697 . . . . . 6 (∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧) ↔ ¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
4 ax-1 6 . . . . . . . . . . 11 (¬ ∀𝑥(𝜑𝑥 = 𝑧) → (𝑧 = 𝑧 → ¬ ∀𝑥(𝜑𝑥 = 𝑧)))
5 eqidd 2611 . . . . . . . . . . 11 (¬ ∀𝑥(𝜑𝑥 = 𝑧) → 𝑧 = 𝑧)
64, 5impbid1 214 . . . . . . . . . 10 (¬ ∀𝑥(𝜑𝑥 = 𝑧) → (𝑧 = 𝑧 ↔ ¬ ∀𝑥(𝜑𝑥 = 𝑧)))
76con2bid 343 . . . . . . . . 9 (¬ ∀𝑥(𝜑𝑥 = 𝑧) → (∀𝑥(𝜑𝑥 = 𝑧) ↔ ¬ 𝑧 = 𝑧))
87alimi 1730 . . . . . . . 8 (∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧) → ∀𝑧(∀𝑥(𝜑𝑥 = 𝑧) ↔ ¬ 𝑧 = 𝑧))
9 abbi 2724 . . . . . . . 8 (∀𝑧(∀𝑥(𝜑𝑥 = 𝑧) ↔ ¬ 𝑧 = 𝑧) ↔ {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = {𝑧 ∣ ¬ 𝑧 = 𝑧})
108, 9sylib 207 . . . . . . 7 (∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = {𝑧 ∣ ¬ 𝑧 = 𝑧})
11 dfnul2 3876 . . . . . . 7 ∅ = {𝑧 ∣ ¬ 𝑧 = 𝑧}
1210, 11syl6eqr 2662 . . . . . 6 (∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = ∅)
133, 12sylbir 224 . . . . 5 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = ∅)
1413unieqd 4382 . . . 4 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = ∅)
15 uni0 4401 . . . 4 ∅ = ∅
1614, 15syl6eq 2660 . . 3 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = ∅)
172, 16syl5eq 2656 . 2 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → (℩𝑥𝜑) = ∅)
181, 17sylnbi 319 1 (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195  ∀wal 1473   = wceq 1475  ∃wex 1695  ∃!weu 2458  {cab 2596  ∅c0 3874  ∪ cuni 4372  ℩cio 5766 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126  df-uni 4373  df-iota 5768 This theorem is referenced by:  iotassuni  5784  iotaex  5785  dfiota4  5796  csbiota  5797  tz6.12-2  6094  dffv3  6099  csbriota  6523  riotaund  6546  isf32lem9  9066  grpidval  17083  0g0  17086
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