ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  iotanul GIF version

Theorem iotanul 4909
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 1919 . . 3 (∃!𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
2 dfiota2 4895 . . . 4 (℩𝑥𝜑) = {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)}
3 alnex 1404 . . . . . . 7 (∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧) ↔ ¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
4 ax-in2 555 . . . . . . . . . 10 (¬ ∀𝑥(𝜑𝑥 = 𝑧) → (∀𝑥(𝜑𝑥 = 𝑧) → ¬ 𝑧 = 𝑧))
54alimi 1360 . . . . . . . . 9 (∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧) → ∀𝑧(∀𝑥(𝜑𝑥 = 𝑧) → ¬ 𝑧 = 𝑧))
6 ss2ab 3035 . . . . . . . . 9 ({𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} ⊆ {𝑧 ∣ ¬ 𝑧 = 𝑧} ↔ ∀𝑧(∀𝑥(𝜑𝑥 = 𝑧) → ¬ 𝑧 = 𝑧))
75, 6sylibr 141 . . . . . . . 8 (∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} ⊆ {𝑧 ∣ ¬ 𝑧 = 𝑧})
8 dfnul2 3253 . . . . . . . 8 ∅ = {𝑧 ∣ ¬ 𝑧 = 𝑧}
97, 8syl6sseqr 3019 . . . . . . 7 (∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} ⊆ ∅)
103, 9sylbir 129 . . . . . 6 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} ⊆ ∅)
1110unissd 3631 . . . . 5 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} ⊆ ∅)
12 uni0 3634 . . . . 5 ∅ = ∅
1311, 12syl6sseq 3018 . . . 4 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} ⊆ ∅)
142, 13syl5eqss 3016 . . 3 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → (℩𝑥𝜑) ⊆ ∅)
151, 14sylnbi 613 . 2 (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) ⊆ ∅)
16 ss0 3284 . 2 ((℩𝑥𝜑) ⊆ ∅ → (℩𝑥𝜑) = ∅)
1715, 16syl 14 1 (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 102  wal 1257   = wceq 1259  wex 1397  ∃!weu 1916  {cab 2042  wss 2944  c0 3251   cuni 3607  cio 4892
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-dif 2947  df-in 2951  df-ss 2958  df-nul 3252  df-sn 3408  df-uni 3608  df-iota 4894
This theorem is referenced by:  tz6.12-2  5196  0fv  5235  riotaund  5529
  Copyright terms: Public domain W3C validator