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Theorem iotanul 4949
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 1946 . . 3 (∃!𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
2 dfiota2 4935 . . . 4 (℩𝑥𝜑) = {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)}
3 alnex 1429 . . . . . . 7 (∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧) ↔ ¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
4 ax-in2 578 . . . . . . . . . 10 (¬ ∀𝑥(𝜑𝑥 = 𝑧) → (∀𝑥(𝜑𝑥 = 𝑧) → ¬ 𝑧 = 𝑧))
54alimi 1385 . . . . . . . . 9 (∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧) → ∀𝑧(∀𝑥(𝜑𝑥 = 𝑧) → ¬ 𝑧 = 𝑧))
6 ss2ab 3073 . . . . . . . . 9 ({𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} ⊆ {𝑧 ∣ ¬ 𝑧 = 𝑧} ↔ ∀𝑧(∀𝑥(𝜑𝑥 = 𝑧) → ¬ 𝑧 = 𝑧))
75, 6sylibr 132 . . . . . . . 8 (∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} ⊆ {𝑧 ∣ ¬ 𝑧 = 𝑧})
8 dfnul2 3271 . . . . . . . 8 ∅ = {𝑧 ∣ ¬ 𝑧 = 𝑧}
97, 8syl6sseqr 3057 . . . . . . 7 (∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} ⊆ ∅)
103, 9sylbir 133 . . . . . 6 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} ⊆ ∅)
1110unissd 3651 . . . . 5 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} ⊆ ∅)
12 uni0 3654 . . . . 5 ∅ = ∅
1311, 12syl6sseq 3056 . . . 4 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} ⊆ ∅)
142, 13syl5eqss 3054 . . 3 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → (℩𝑥𝜑) ⊆ ∅)
151, 14sylnbi 636 . 2 (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) ⊆ ∅)
16 ss0 3305 . 2 ((℩𝑥𝜑) ⊆ ∅ → (℩𝑥𝜑) = ∅)
1715, 16syl 14 1 (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 103  wal 1283   = wceq 1285  wex 1422  ∃!weu 1943  {cab 2069  wss 2984  c0 3269   cuni 3627  cio 4932
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-dif 2986  df-in 2990  df-ss 2997  df-nul 3270  df-sn 3428  df-uni 3628  df-iota 4934
This theorem is referenced by:  tz6.12-2  5244  0fv  5284  riotaund  5581
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