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Theorem iotanul 6538
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 2573 . 2 (∃!𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
2 dfiota2 6514 . . 3 (℩𝑥𝜑) = {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)}
3 alnex 1780 . . . . . 6 (∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧) ↔ ¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
4 dfnul2 4335 . . . . . . 7 ∅ = {𝑧 ∣ ¬ 𝑧 = 𝑧}
5 equid 2010 . . . . . . . . . . . 12 𝑧 = 𝑧
65tbt 369 . . . . . . . . . . 11 (¬ ∀𝑥(𝜑𝑥 = 𝑧) ↔ (¬ ∀𝑥(𝜑𝑥 = 𝑧) ↔ 𝑧 = 𝑧))
76biimpi 216 . . . . . . . . . 10 (¬ ∀𝑥(𝜑𝑥 = 𝑧) → (¬ ∀𝑥(𝜑𝑥 = 𝑧) ↔ 𝑧 = 𝑧))
87con1bid 355 . . . . . . . . 9 (¬ ∀𝑥(𝜑𝑥 = 𝑧) → (¬ 𝑧 = 𝑧 ↔ ∀𝑥(𝜑𝑥 = 𝑧)))
98alimi 1810 . . . . . . . 8 (∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧) → ∀𝑧𝑧 = 𝑧 ↔ ∀𝑥(𝜑𝑥 = 𝑧)))
10 abbi 2806 . . . . . . . 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 2789 . . . . . 6 (∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = ∅)
133, 12sylbir 235 . . . . 5 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = ∅)
1413unieqd 4919 . . . 4 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = ∅)
15 uni0 4934 . . . 4 ∅ = ∅
1614, 15eqtrdi 2792 . . 3 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = ∅)
172, 16eqtrid 2788 . 2 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → (℩𝑥𝜑) = ∅)
181, 17sylnbi 330 1 (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wal 1537   = wceq 1539  wex 1778  ∃!weu 2567  {cab 2713  c0 4332   cuni 4906  cio 6511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-v 3481  df-dif 3953  df-ss 3967  df-nul 4333  df-sn 4626  df-uni 4907  df-iota 6513
This theorem is referenced by:  iotassuniOLD  6539  iotaexOLD  6540  iotan0  6550  dfiota4  6552  csbiota  6553  tz6.12-2  6893  dffv3  6901  csbriota  7404  riotaund  7428  isf32lem9  10402  grpidval  18675  0g0  18678  iota0ndef  47056  iotan0aiotaex  47110
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