MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iotanul Structured version   Visualization version   GIF version

Theorem iotanul 5765
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 2457 . 2 (∃!𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
2 dfiota2 5751 . . 3 (℩𝑥𝜑) = {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)}
3 alnex 1696 . . . . . 6 (∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧) ↔ ¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
4 ax-1 6 . . . . . . . . . . 11 (¬ ∀𝑥(𝜑𝑥 = 𝑧) → (𝑧 = 𝑧 → ¬ ∀𝑥(𝜑𝑥 = 𝑧)))
5 eqidd 2606 . . . . . . . . . . 11 (¬ ∀𝑥(𝜑𝑥 = 𝑧) → 𝑧 = 𝑧)
64, 5impbid1 213 . . . . . . . . . 10 (¬ ∀𝑥(𝜑𝑥 = 𝑧) → (𝑧 = 𝑧 ↔ ¬ ∀𝑥(𝜑𝑥 = 𝑧)))
76con2bid 342 . . . . . . . . 9 (¬ ∀𝑥(𝜑𝑥 = 𝑧) → (∀𝑥(𝜑𝑥 = 𝑧) ↔ ¬ 𝑧 = 𝑧))
87alimi 1728 . . . . . . . 8 (∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧) → ∀𝑧(∀𝑥(𝜑𝑥 = 𝑧) ↔ ¬ 𝑧 = 𝑧))
9 abbi 2719 . . . . . . . 8 (∀𝑧(∀𝑥(𝜑𝑥 = 𝑧) ↔ ¬ 𝑧 = 𝑧) ↔ {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = {𝑧 ∣ ¬ 𝑧 = 𝑧})
108, 9sylib 206 . . . . . . 7 (∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = {𝑧 ∣ ¬ 𝑧 = 𝑧})
11 dfnul2 3871 . . . . . . 7 ∅ = {𝑧 ∣ ¬ 𝑧 = 𝑧}
1210, 11syl6eqr 2657 . . . . . 6 (∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = ∅)
133, 12sylbir 223 . . . . 5 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = ∅)
1413unieqd 4372 . . . 4 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = ∅)
15 uni0 4391 . . . 4 ∅ = ∅
1614, 15syl6eq 2655 . . 3 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = ∅)
172, 16syl5eq 2651 . 2 (¬ ∃𝑧𝑥(𝜑𝑥 = 𝑧) → (℩𝑥𝜑) = ∅)
181, 17sylnbi 318 1 (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wal 1472   = wceq 1474  wex 1694  ∃!weu 2453  {cab 2591  c0 3869   cuni 4362  cio 5748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-v 3170  df-dif 3538  df-in 3542  df-ss 3549  df-nul 3870  df-sn 4121  df-uni 4363  df-iota 5750
This theorem is referenced by:  iotassuni  5766  iotaex  5767  dfiota4  5778  csbiota  5779  tz6.12-2  6075  dffv3  6080  csbriota  6497  riotaund  6520  isf32lem9  9039  grpidval  17025  0g0  17028
  Copyright terms: Public domain W3C validator