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Theorem iotanul 4882
 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 1903 . . 3 (∃!𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
2 dfiota2 4868 . . . 4 (℩𝑥𝜑) = {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)}
3 alnex 1388 . . . . . . 7 (∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧) ↔ ¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
4 ax-in2 545 . . . . . . . . . 10 (¬ ∀𝑥(𝜑𝑥 = 𝑧) → (∀𝑥(𝜑𝑥 = 𝑧) → ¬ 𝑧 = 𝑧))
54alimi 1344 . . . . . . . . 9 (∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧) → ∀𝑧(∀𝑥(𝜑𝑥 = 𝑧) → ¬ 𝑧 = 𝑧))
6 ss2ab 3008 . . . . . . . . 9 ({𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} ⊆ {𝑧 ∣ ¬ 𝑧 = 𝑧} ↔ ∀𝑧(∀𝑥(𝜑𝑥 = 𝑧) → ¬ 𝑧 = 𝑧))
75, 6sylibr 137 . . . . . . . 8 (∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} ⊆ {𝑧 ∣ ¬ 𝑧 = 𝑧})
8 dfnul2 3226 . . . . . . . 8 ∅ = {𝑧 ∣ ¬ 𝑧 = 𝑧}
97, 8syl6sseqr 2992 . . . . . . 7 (∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} ⊆ ∅)
103, 9sylbir 125 . . . . . 6 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} ⊆ ∅)
1110unissd 3604 . . . . 5 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} ⊆ ∅)
12 uni0 3607 . . . . 5 ∅ = ∅
1311, 12syl6sseq 2991 . . . 4 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} ⊆ ∅)
142, 13syl5eqss 2989 . . 3 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → (℩𝑥𝜑) ⊆ ∅)
151, 14sylnbi 603 . 2 (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) ⊆ ∅)
16 ss0 3257 . 2 ((℩𝑥𝜑) ⊆ ∅ → (℩𝑥𝜑) = ∅)
1715, 16syl 14 1 (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98  ∀wal 1241   = wceq 1243  ∃wex 1381  ∃!weu 1900  {cab 2026   ⊆ wss 2917  ∅c0 3224  ∪ cuni 3580  ℩cio 4865 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-in 2924  df-ss 2931  df-nul 3225  df-sn 3381  df-uni 3581  df-iota 4867 This theorem is referenced by:  tz6.12-2  5169  0fv  5208  riotaund  5502
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