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Theorem iotanul 6080
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 2614 . 2 (∃!𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
2 dfiota2 6066 . . 3 (℩𝑥𝜑) = {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)}
3 alnex 1877 . . . . . 6 (∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧) ↔ ¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
4 dfnul2 4118 . . . . . . 7 ∅ = {𝑧 ∣ ¬ 𝑧 = 𝑧}
5 equid 2111 . . . . . . . . . . . 12 𝑧 = 𝑧
65tbt 361 . . . . . . . . . . 11 (¬ ∀𝑥(𝜑𝑥 = 𝑧) ↔ (¬ ∀𝑥(𝜑𝑥 = 𝑧) ↔ 𝑧 = 𝑧))
76biimpi 208 . . . . . . . . . 10 (¬ ∀𝑥(𝜑𝑥 = 𝑧) → (¬ ∀𝑥(𝜑𝑥 = 𝑧) ↔ 𝑧 = 𝑧))
87con1bid 347 . . . . . . . . 9 (¬ ∀𝑥(𝜑𝑥 = 𝑧) → (¬ 𝑧 = 𝑧 ↔ ∀𝑥(𝜑𝑥 = 𝑧)))
98alimi 1907 . . . . . . . 8 (∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧) → ∀𝑧𝑧 = 𝑧 ↔ ∀𝑥(𝜑𝑥 = 𝑧)))
10 abbi 2915 . . . . . . . 8 (∀𝑧𝑧 = 𝑧 ↔ ∀𝑥(𝜑𝑥 = 𝑧)) ↔ {𝑧 ∣ ¬ 𝑧 = 𝑧} = {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)})
119, 10sylib 210 . . . . . . 7 (∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ¬ 𝑧 = 𝑧} = {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)})
124, 11syl5req 2847 . . . . . 6 (∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = ∅)
133, 12sylbir 227 . . . . 5 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = ∅)
1413unieqd 4639 . . . 4 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = ∅)
15 uni0 4658 . . . 4 ∅ = ∅
1614, 15syl6eq 2850 . . 3 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = ∅)
172, 16syl5eq 2846 . 2 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → (℩𝑥𝜑) = ∅)
181, 17sylnbi 322 1 (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wal 1651   = wceq 1653  wex 1875  ∃!weu 2609  {cab 2786  c0 4116   cuni 4629  cio 6063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ral 3095  df-rex 3096  df-v 3388  df-dif 3773  df-in 3777  df-ss 3784  df-nul 4117  df-sn 4370  df-uni 4630  df-iota 6065
This theorem is referenced by:  iotassuni  6081  iotaex  6082  dfiota4  6093  dfiota4OLD  6094  csbiota  6095  tz6.12-2  6402  dffv3  6408  csbriota  6852  riotaund  6876  isf32lem9  9472  grpidval  17574  0g0  17577  iota0ndef  41917  iotan0aiotaex  41935
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