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Type | Label | Description |
---|---|---|
Statement | ||
Syntax | wmo 2001 | Extend wff definition to include uniqueness ("there exists at most one 𝑥 such that 𝜑"). |
wff ∃*𝑥𝜑 | ||
Theorem | eujust 2002* | A soundness justification theorem for df-eu 2003, showing that the definition is equivalent to itself with its dummy variable renamed. Note that 𝑦 and 𝑧 needn't be distinct variables. (Contributed by NM, 11-Mar-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧)) | ||
Definition | df-eu 2003* | Define existential uniqueness, i.e. "there exists exactly one 𝑥 such that 𝜑." Definition 10.1 of [BellMachover] p. 97; also Definition *14.02 of [WhiteheadRussell] p. 175. Other possible definitions are given by eu1 2025, eu2 2044, eu3 2046, and eu5 2047 (which in some cases we show with a hypothesis 𝜑 → ∀𝑦𝜑 in place of a distinct variable condition on 𝑦 and 𝜑). Double uniqueness is tricky: ∃!𝑥∃!𝑦𝜑 does not mean "exactly one 𝑥 and one 𝑦 " (see 2eu4 2093). (Contributed by NM, 5-Aug-1993.) |
⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | ||
Definition | df-mo 2004 | Define "there exists at most one 𝑥 such that 𝜑." Here we define it in terms of existential uniqueness. Notation of [BellMachover] p. 460, whose definition we show as mo3 2054. For another possible definition see mo4 2061. (Contributed by NM, 5-Aug-1993.) |
⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) | ||
Theorem | euf 2005* | A version of the existential uniqueness definition with a hypothesis instead of a distinct variable condition. (Contributed by NM, 12-Aug-1993.) |
⊢ (𝜑 → ∀𝑦𝜑) ⇒ ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | ||
Theorem | eubidh 2006 | Formula-building rule for unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒)) | ||
Theorem | eubid 2007 | Formula-building rule for unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒)) | ||
Theorem | eubidv 2008* | Formula-building rule for unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒)) | ||
Theorem | eubii 2009 | Introduce unique existential quantifier to both sides of an equivalence. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 6-Oct-2016.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (∃!𝑥𝜑 ↔ ∃!𝑥𝜓) | ||
Theorem | hbeu1 2010 | Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.) |
⊢ (∃!𝑥𝜑 → ∀𝑥∃!𝑥𝜑) | ||
Theorem | nfeu1 2011 | Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) |
⊢ Ⅎ𝑥∃!𝑥𝜑 | ||
Theorem | nfmo1 2012 | Bound-variable hypothesis builder for "at most one." (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.) |
⊢ Ⅎ𝑥∃*𝑥𝜑 | ||
Theorem | sb8eu 2013 | Variable substitution in unique existential quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑) | ||
Theorem | sb8mo 2014 | Variable substitution for "at most one." (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑) | ||
Theorem | nfeudv 2015* | Deduction version of nfeu 2019. Similar to nfeud 2016 but has the additional constraint that 𝑥 and 𝑦 must be distinct. (Contributed by Jim Kingdon, 25-May-2018.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓) | ||
Theorem | nfeud 2016 | Deduction version of nfeu 2019. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof rewritten by Jim Kingdon, 25-May-2018.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓) | ||
Theorem | nfmod 2017 | Bound-variable hypothesis builder for "at most one." (Contributed by Mario Carneiro, 14-Nov-2016.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓) | ||
Theorem | nfeuv 2018* | Bound-variable hypothesis builder for existential uniqueness. This is similar to nfeu 2019 but has the additional constraint that 𝑥 and 𝑦 must be distinct. (Contributed by Jim Kingdon, 23-May-2018.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥∃!𝑦𝜑 | ||
Theorem | nfeu 2019 | Bound-variable hypothesis builder for existential uniqueness. Note that 𝑥 and 𝑦 needn't be distinct. (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof rewritten by Jim Kingdon, 23-May-2018.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥∃!𝑦𝜑 | ||
Theorem | nfmo 2020 | Bound-variable hypothesis builder for "at most one." (Contributed by NM, 9-Mar-1995.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥∃*𝑦𝜑 | ||
Theorem | hbeu 2021 | Bound-variable hypothesis builder for uniqueness. Note that 𝑥 and 𝑦 needn't be distinct. (Contributed by NM, 8-Mar-1995.) (Proof rewritten by Jim Kingdon, 24-May-2018.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ (∃!𝑦𝜑 → ∀𝑥∃!𝑦𝜑) | ||
Theorem | hbeud 2022 | Deduction version of hbeu 2021. (Contributed by NM, 15-Feb-2013.) (Proof rewritten by Jim Kingdon, 25-May-2018.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → ∀𝑦𝜑) & ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) ⇒ ⊢ (𝜑 → (∃!𝑦𝜓 → ∀𝑥∃!𝑦𝜓)) | ||
Theorem | sb8euh 2023 | Variable substitution in unique existential quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Andrew Salmon, 9-Jul-2011.) |
⊢ (𝜑 → ∀𝑦𝜑) ⇒ ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑) | ||
Theorem | cbveu 2024 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓) | ||
Theorem | eu1 2025* | An alternate way to express uniqueness used by some authors. Exercise 2(b) of [Margaris] p. 110. (Contributed by NM, 20-Aug-1993.) |
⊢ (𝜑 → ∀𝑦𝜑) ⇒ ⊢ (∃!𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦))) | ||
Theorem | euor 2026 | Introduce a disjunct into a unique existential quantifier. (Contributed by NM, 21-Oct-2005.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ ((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑 ∨ 𝜓)) | ||
Theorem | euorv 2027* | Introduce a disjunct into a unique existential quantifier. (Contributed by NM, 23-Mar-1995.) |
⊢ ((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑 ∨ 𝜓)) | ||
Theorem | mo2n 2028* | There is at most one of something which does not exist. (Contributed by Jim Kingdon, 2-Jul-2018.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (¬ ∃𝑥𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) | ||
Theorem | mon 2029 | There is at most one of something which does not exist. (Contributed by Jim Kingdon, 5-Jul-2018.) |
⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑) | ||
Theorem | euex 2030 | Existential uniqueness implies existence. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
⊢ (∃!𝑥𝜑 → ∃𝑥𝜑) | ||
Theorem | eumo0 2031* | Existential uniqueness implies "at most one." (Contributed by NM, 8-Jul-1994.) |
⊢ (𝜑 → ∀𝑦𝜑) ⇒ ⊢ (∃!𝑥𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) | ||
Theorem | eumo 2032 | Existential uniqueness implies "at most one." (Contributed by NM, 23-Mar-1995.) (Proof rewritten by Jim Kingdon, 27-May-2018.) |
⊢ (∃!𝑥𝜑 → ∃*𝑥𝜑) | ||
Theorem | eumoi 2033 | "At most one" inferred from existential uniqueness. (Contributed by NM, 5-Apr-1995.) |
⊢ ∃!𝑥𝜑 ⇒ ⊢ ∃*𝑥𝜑 | ||
Theorem | mobidh 2034 | Formula-building rule for "at most one" quantifier (deduction form). (Contributed by NM, 8-Mar-1995.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)) | ||
Theorem | mobid 2035 | Formula-building rule for "at most one" quantifier (deduction form). (Contributed by NM, 8-Mar-1995.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)) | ||
Theorem | mobidv 2036* | Formula-building rule for "at most one" quantifier (deduction form). (Contributed by Mario Carneiro, 7-Oct-2016.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)) | ||
Theorem | mobii 2037 | Formula-building rule for "at most one" quantifier (inference form). (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 17-Oct-2016.) |
⊢ (𝜓 ↔ 𝜒) ⇒ ⊢ (∃*𝑥𝜓 ↔ ∃*𝑥𝜒) | ||
Theorem | hbmo1 2038 | Bound-variable hypothesis builder for "at most one." (Contributed by NM, 8-Mar-1995.) |
⊢ (∃*𝑥𝜑 → ∀𝑥∃*𝑥𝜑) | ||
Theorem | hbmo 2039 | Bound-variable hypothesis builder for "at most one." (Contributed by NM, 9-Mar-1995.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ (∃*𝑦𝜑 → ∀𝑥∃*𝑦𝜑) | ||
Theorem | cbvmo 2040 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon, 8-Jun-2011.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓) | ||
Theorem | mo23 2041* | An implication between two definitions of "there exists at most one." (Contributed by Jim Kingdon, 25-Jun-2018.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) | ||
Theorem | mor 2042* | Converse of mo23 2041 with an additional ∃𝑥𝜑 condition. (Contributed by Jim Kingdon, 25-Jun-2018.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃𝑥𝜑 → (∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) | ||
Theorem | modc 2043* | Equivalent definitions of "there exists at most one," given decidable existence. (Contributed by Jim Kingdon, 1-Jul-2018.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (DECID ∃𝑥𝜑 → (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))) | ||
Theorem | eu2 2044* | An alternate way of defining existential uniqueness. Definition 6.10 of [TakeutiZaring] p. 26. (Contributed by NM, 8-Jul-1994.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))) | ||
Theorem | eu3h 2045* | An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) (New usage is discouraged.) |
⊢ (𝜑 → ∀𝑦𝜑) ⇒ ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) | ||
Theorem | eu3 2046* | An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) | ||
Theorem | eu5 2047 | Uniqueness in terms of "at most one." (Contributed by NM, 23-Mar-1995.) (Proof rewritten by Jim Kingdon, 27-May-2018.) |
⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑)) | ||
Theorem | exmoeu2 2048 | Existence implies "at most one" is equivalent to uniqueness. (Contributed by NM, 5-Apr-2004.) |
⊢ (∃𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃!𝑥𝜑)) | ||
Theorem | moabs 2049 | Absorption of existence condition by "at most one." (Contributed by NM, 4-Nov-2002.) |
⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑)) | ||
Theorem | exmodc 2050 | If existence is decidable, something exists or at most one exists. (Contributed by Jim Kingdon, 30-Jun-2018.) |
⊢ (DECID ∃𝑥𝜑 → (∃𝑥𝜑 ∨ ∃*𝑥𝜑)) | ||
Theorem | exmonim 2051 | There is at most one of something which does not exist. Unlike exmodc 2050 there is no decidability condition. (Contributed by Jim Kingdon, 22-Sep-2018.) |
⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑) | ||
Theorem | mo2r 2052* | A condition which implies "at most one." (Contributed by Jim Kingdon, 2-Jul-2018.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃*𝑥𝜑) | ||
Theorem | mo3h 2053* | Alternate definition of "at most one." Definition of [BellMachover] p. 460, except that definition has the side condition that 𝑦 not occur in 𝜑 in place of our hypothesis. (Contributed by NM, 8-Mar-1995.) (New usage is discouraged.) |
⊢ (𝜑 → ∀𝑦𝜑) ⇒ ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) | ||
Theorem | mo3 2054* | Alternate definition of "at most one." Definition of [BellMachover] p. 460, except that definition has the side condition that 𝑦 not occur in 𝜑 in place of our hypothesis. (Contributed by NM, 8-Mar-1995.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) | ||
Theorem | mo2dc 2055* | Alternate definition of "at most one" where existence is decidable. (Contributed by Jim Kingdon, 2-Jul-2018.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (DECID ∃𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) | ||
Theorem | euan 2056 | Introduction of a conjunct into unique existential quantifier. (Contributed by NM, 19-Feb-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ (∃!𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓)) | ||
Theorem | euanv 2057* | Introduction of a conjunct into unique existential quantifier. (Contributed by NM, 23-Mar-1995.) |
⊢ (∃!𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓)) | ||
Theorem | euor2 2058 | Introduce or eliminate a disjunct in a unique existential quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
⊢ (¬ ∃𝑥𝜑 → (∃!𝑥(𝜑 ∨ 𝜓) ↔ ∃!𝑥𝜓)) | ||
Theorem | sbmo 2059* | Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ([𝑦 / 𝑥]∃*𝑧𝜑 ↔ ∃*𝑧[𝑦 / 𝑥]𝜑) | ||
Theorem | mo4f 2060* | "At most one" expressed using implicit substitution. (Contributed by NM, 10-Apr-2004.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) | ||
Theorem | mo4 2061* | "At most one" expressed using implicit substitution. (Contributed by NM, 26-Jul-1995.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) | ||
Theorem | eu4 2062* | Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))) | ||
Theorem | exmoeudc 2063 | Existence in terms of "at most one" and uniqueness. (Contributed by Jim Kingdon, 3-Jul-2018.) |
⊢ (DECID ∃𝑥𝜑 → (∃𝑥𝜑 ↔ (∃*𝑥𝜑 → ∃!𝑥𝜑))) | ||
Theorem | moim 2064 | "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 22-Apr-1995.) |
⊢ (∀𝑥(𝜑 → 𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑)) | ||
Theorem | moimi 2065 | "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 15-Feb-2006.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (∃*𝑥𝜓 → ∃*𝑥𝜑) | ||
Theorem | moimv 2066* | Move antecedent outside of "at most one." (Contributed by NM, 28-Jul-1995.) |
⊢ (∃*𝑥(𝜑 → 𝜓) → (𝜑 → ∃*𝑥𝜓)) | ||
Theorem | euimmo 2067 | Uniqueness implies "at most one" through implication. (Contributed by NM, 22-Apr-1995.) |
⊢ (∀𝑥(𝜑 → 𝜓) → (∃!𝑥𝜓 → ∃*𝑥𝜑)) | ||
Theorem | euim 2068 | Add existential unique existential quantifiers to an implication. Note the reversed implication in the antecedent. (Contributed by NM, 19-Oct-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) |
⊢ ((∃𝑥𝜑 ∧ ∀𝑥(𝜑 → 𝜓)) → (∃!𝑥𝜓 → ∃!𝑥𝜑)) | ||
Theorem | moan 2069 | "At most one" is still the case when a conjunct is added. (Contributed by NM, 22-Apr-1995.) |
⊢ (∃*𝑥𝜑 → ∃*𝑥(𝜓 ∧ 𝜑)) | ||
Theorem | moani 2070 | "At most one" is still true when a conjunct is added. (Contributed by NM, 9-Mar-1995.) |
⊢ ∃*𝑥𝜑 ⇒ ⊢ ∃*𝑥(𝜓 ∧ 𝜑) | ||
Theorem | moor 2071 | "At most one" is still the case when a disjunct is removed. (Contributed by NM, 5-Apr-2004.) |
⊢ (∃*𝑥(𝜑 ∨ 𝜓) → ∃*𝑥𝜑) | ||
Theorem | mooran1 2072 | "At most one" imports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
⊢ ((∃*𝑥𝜑 ∨ ∃*𝑥𝜓) → ∃*𝑥(𝜑 ∧ 𝜓)) | ||
Theorem | mooran2 2073 | "At most one" exports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
⊢ (∃*𝑥(𝜑 ∨ 𝜓) → (∃*𝑥𝜑 ∧ ∃*𝑥𝜓)) | ||
Theorem | moanim 2074 | Introduction of a conjunct into at-most-one quantifier. (Contributed by NM, 3-Dec-2001.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (∃*𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥𝜓)) | ||
Theorem | moanimv 2075* | Introduction of a conjunct into at-most-one quantifier. (Contributed by NM, 23-Mar-1995.) |
⊢ (∃*𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥𝜓)) | ||
Theorem | moaneu 2076 | Nested at-most-one and unique existential quantifiers. (Contributed by NM, 25-Jan-2006.) |
⊢ ∃*𝑥(𝜑 ∧ ∃!𝑥𝜑) | ||
Theorem | moanmo 2077 | Nested at-most-one quantifiers. (Contributed by NM, 25-Jan-2006.) |
⊢ ∃*𝑥(𝜑 ∧ ∃*𝑥𝜑) | ||
Theorem | mopick 2078 | "At most one" picks a variable value, eliminating an existential quantifier. (Contributed by NM, 27-Jan-1997.) |
⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓)) | ||
Theorem | eupick 2079 | Existential uniqueness "picks" a variable value for which another wff is true. If there is only one thing 𝑥 such that 𝜑 is true, and there is also an 𝑥 (actually the same one) such that 𝜑 and 𝜓 are both true, then 𝜑 implies 𝜓 regardless of 𝑥. This theorem can be useful for eliminating existential quantifiers in a hypothesis. Compare Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by NM, 10-Jul-1994.) |
⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓)) | ||
Theorem | eupicka 2080 | Version of eupick 2079 with closed formulas. (Contributed by NM, 6-Sep-2008.) |
⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → ∀𝑥(𝜑 → 𝜓)) | ||
Theorem | eupickb 2081 | Existential uniqueness "pick" showing wff equivalence. (Contributed by NM, 25-Nov-1994.) |
⊢ ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 ↔ 𝜓)) | ||
Theorem | eupickbi 2082 | Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.) |
⊢ (∃!𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) ↔ ∀𝑥(𝜑 → 𝜓))) | ||
Theorem | mopick2 2083 | "At most one" can show the existence of a common value. In this case we can infer existence of conjunction from a conjunction of existence, and it is one way to achieve the converse of 19.40 1611. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓) ∧ ∃𝑥(𝜑 ∧ 𝜒)) → ∃𝑥(𝜑 ∧ 𝜓 ∧ 𝜒)) | ||
Theorem | moexexdc 2084 | "At most one" double quantification. (Contributed by Jim Kingdon, 5-Jul-2018.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (DECID ∃𝑥𝜑 → ((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦∃𝑥(𝜑 ∧ 𝜓))) | ||
Theorem | euexex 2085 | Existential uniqueness and "at most one" double quantification. (Contributed by Jim Kingdon, 28-Dec-2018.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ ((∃!𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦∃𝑥(𝜑 ∧ 𝜓)) | ||
Theorem | 2moex 2086 | Double quantification with "at most one." (Contributed by NM, 3-Dec-2001.) |
⊢ (∃*𝑥∃𝑦𝜑 → ∀𝑦∃*𝑥𝜑) | ||
Theorem | 2euex 2087 | Double quantification with existential uniqueness. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
⊢ (∃!𝑥∃𝑦𝜑 → ∃𝑦∃!𝑥𝜑) | ||
Theorem | 2eumo 2088 | Double quantification with existential uniqueness and "at most one." (Contributed by NM, 3-Dec-2001.) |
⊢ (∃!𝑥∃*𝑦𝜑 → ∃*𝑥∃!𝑦𝜑) | ||
Theorem | 2eu2ex 2089 | Double existential uniqueness. (Contributed by NM, 3-Dec-2001.) |
⊢ (∃!𝑥∃!𝑦𝜑 → ∃𝑥∃𝑦𝜑) | ||
Theorem | 2moswapdc 2090 | A condition allowing swap of "at most one" and existential quantifiers. (Contributed by Jim Kingdon, 6-Jul-2018.) |
⊢ (DECID ∃𝑥∃𝑦𝜑 → (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥𝜑))) | ||
Theorem | 2euswapdc 2091 | A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Jim Kingdon, 7-Jul-2018.) |
⊢ (DECID ∃𝑥∃𝑦𝜑 → (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃𝑦𝜑 → ∃!𝑦∃𝑥𝜑))) | ||
Theorem | 2exeu 2092 | Double existential uniqueness implies double unique existential quantification. (Contributed by NM, 3-Dec-2001.) |
⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → ∃!𝑥∃!𝑦𝜑) | ||
Theorem | 2eu4 2093* | This theorem provides us with a definition of double existential uniqueness ("exactly one 𝑥 and exactly one 𝑦"). Naively one might think (incorrectly) that it could be defined by ∃!𝑥∃!𝑦𝜑. See 2exeu 2092 for a one-way implication. (Contributed by NM, 3-Dec-2001.) |
⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))) | ||
Theorem | 2eu7 2094 | Two equivalent expressions for double existential uniqueness. (Contributed by NM, 19-Feb-2005.) |
⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ ∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑)) | ||
Theorem | euequ1 2095* | Equality has existential uniqueness. (Contributed by Stefan Allan, 4-Dec-2008.) |
⊢ ∃!𝑥 𝑥 = 𝑦 | ||
Theorem | exists1 2096* | Two ways to express "only one thing exists." The left-hand side requires only one variable to express this. Both sides are false in set theory. (Contributed by NM, 5-Apr-2004.) |
⊢ (∃!𝑥 𝑥 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦) | ||
Theorem | exists2 2097 | A condition implying that at least two things exist. (Contributed by NM, 10-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
⊢ ((∃𝑥𝜑 ∧ ∃𝑥 ¬ 𝜑) → ¬ ∃!𝑥 𝑥 = 𝑥) | ||
Model the Aristotelian assertic syllogisms using modern notation. This section shows that the Aristotelian assertic syllogisms can be proven with our axioms of logic, and also provides generally useful theorems. In antiquity Aristotelian logic and Stoic logic (see mptnan 1402) were the leading logical systems. Aristotelian logic became the leading system in medieval Europe; this section models this system (including later refinements to it). Aristotle defined syllogisms very generally ("a discourse in which certain (specific) things having been supposed, something different from the things supposed results of necessity because these things are so") Aristotle, Prior Analytics 24b18-20. However, in Prior Analytics he limits himself to categorical syllogisms that consist of three categorical propositions with specific structures. The syllogisms are the valid subset of the possible combinations of these structures. The medieval schools used vowels to identify the types of terms (a=all, e=none, i=some, and o=some are not), and named the different syllogisms with Latin words that had the vowels in the intended order. "There is a surprising amount of scholarly debate about how best to formalize Aristotle's syllogisms..." according to Aristotle's Modal Proofs: Prior Analytics A8-22 in Predicate Logic, Adriane Rini, Springer, 2011, ISBN 978-94-007-0049-9, page 28. For example, Lukasiewicz believes it is important to note that "Aristotle does not introduce singular terms or premisses into his system". Lukasiewicz also believes that Aristotelian syllogisms are predicates (having a true/false value), not inference rules: "The characteristic sign of an inference is the word 'therefore'... no syllogism is formulated by Aristotle primarily as an inference, but they are all implications." Jan Lukasiewicz, Aristotle's Syllogistic from the Standpoint of Modern Formal Logic, Second edition, Oxford, 1957, page 1-2. Lukasiewicz devised a specialized prefix notation for representing Aristotelian syllogisms instead of using standard predicate logic notation. We instead translate each Aristotelian syllogism into an inference rule, and each rule is defined using standard predicate logic notation and predicates. The predicates are represented by wff variables that may depend on the quantified variable 𝑥. Our translation is essentially identical to the one use in Rini page 18, Table 2 "Non-Modal Syllogisms in Lower Predicate Calculus (LPC)", which uses standard predicate logic with predicates. Rini states, "the crucial point is that we capture the meaning Aristotle intends, and the method by which we represent that meaning is less important." There are two differences: we make the existence criteria explicit, and we use 𝜑, 𝜓, and 𝜒 in the order they appear (a common Metamath convention). Patzig also uses standard predicate logic notation and predicates (though he interprets them as conditional propositions, not as inference rules); see Gunther Patzig, Aristotle's Theory of the Syllogism second edition, 1963, English translation by Jonathan Barnes, 1968, page 38. Terms such as "all" and "some" are translated into predicate logic using the aproach devised by Frege and Russell. "Frege (and Russell) devised an ingenious procedure for regimenting binary quantifiers like "every" and "some" in terms of unary quantifiers like "everything" and "something": they formalized sentences of the form "Some A is B" and "Every A is B" as exists x (Ax and Bx) and all x (Ax implies Bx), respectively." "Quantifiers and Quantification", Stanford Encyclopedia of Philosophy, http://plato.stanford.edu/entries/quantification/ 1402. See Principia Mathematica page 22 and *10 for more information (especially *10.3 and *10.26). Expressions of the form "no 𝜑 is 𝜓 " are consistently translated as ∀𝑥(𝜑 → ¬ 𝜓). These can also be expressed as ¬ ∃𝑥(𝜑 ∧ 𝜓), per alinexa 1583. We translate "all 𝜑 is 𝜓 " to ∀𝑥(𝜑 → 𝜓), "some 𝜑 is 𝜓 " to ∃𝑥(𝜑 ∧ 𝜓), and "some 𝜑 is not 𝜓 " to ∃𝑥(𝜑 ∧ ¬ 𝜓). It is traditional to use the singular verb "is", not the plural verb "are", in the generic expressions. By convention the major premise is listed first. In traditional Aristotelian syllogisms the predicates have a restricted form ("x is a ..."); those predicates could be modeled in modern notation by constructs such as 𝑥 = 𝐴, 𝑥 ∈ 𝐴, or 𝑥 ⊆ 𝐴. Here we use wff variables instead of specialized restricted forms. This generalization makes the syllogisms more useful in more circumstances. In addition, these expressions make it clearer that the syllogisms of Aristolean logic are the forerunners of predicate calculus. If we used restricted forms like 𝑥 ∈ 𝐴 instead, we would not only unnecessarily limit their use, but we would also need to use set and class axioms, making their relationship to predicate calculus less clear. There are some widespread misconceptions about the existential assumptions made by Aristotle (aka "existential import"). Aristotle was not trying to develop something exactly corresponding to modern logic. Aristotle devised "a companion-logic for science. He relegates fictions like fairy godmothers and mermaids and unicorns to the realms of poetry and literature. In his mind, they exist outside the ambit of science. This is why he leaves no room for such nonexistent entities in his logic. This is a thoughtful choice, not an inadvertent omission. Technically, Aristotelian science is a search for definitions, where a definition is "a phrase signifying a thing's essence." (Topics, I.5.102a37, Pickard-Cambridge.)... Because nonexistent entities cannot be anything, they do not, in Aristotle's mind, possess an essence... This is why he leaves no place for fictional entities like goat-stags (or unicorns)." Source: Louis F. Groarke, "Aristotle: Logic", section 7. (Existential Assumptions), Internet Encyclopedia of Philosophy (A Peer-Reviewed Academic Resource), http://www.iep.utm.edu/aris-log/ 1583. Thus, some syllogisms have "extra" existence hypotheses that do not directly appear in Aristotle's original materials (since they were always assumed); they are added where they are needed. This affects barbari 2102, celaront 2103, cesaro 2108, camestros 2109, felapton 2114, darapti 2115, calemos 2119, fesapo 2120, and bamalip 2121. These are only the assertic syllogisms. Aristotle also defined modal syllogisms that deal with modal qualifiers such as "necessarily" and "possibly". Historically Aristotelian modal syllogisms were not as widely used. For more about modal syllogisms in a modern context, see Rini as well as Aristotle's Modal Syllogistic by Marko Malink, Harvard University Press, November 2013. We do not treat them further here. Aristotelean logic is essentially the forerunner of predicate calculus (as well as set theory since it discusses membership in groups), while Stoic logic is essentially the forerunner of propositional calculus. | ||
Theorem | barbara 2098 | "Barbara", one of the fundamental syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and all 𝜒 is 𝜑, therefore all 𝜒 is 𝜓. (In Aristotelian notation, AAA-1: MaP and SaM therefore SaP.) For example, given "All men are mortal" and "Socrates is a man", we can prove "Socrates is mortal". If H is the set of men, M is the set of mortal beings, and S is Socrates, these word phrases can be represented as ∀𝑥(𝑥 ∈ 𝐻 → 𝑥 ∈ 𝑀) (all men are mortal) and ∀𝑥(𝑥 = 𝑆 → 𝑥 ∈ 𝐻) (Socrates is a man) therefore ∀𝑥(𝑥 = 𝑆 → 𝑥 ∈ 𝑀) (Socrates is mortal). Russell and Whitehead note that the "syllogism in Barbara is derived..." from syl 14. (quote after Theorem *2.06 of [WhiteheadRussell] p. 101). Most of the proof is in alsyl 1615. There are a legion of sources for Barbara, including http://www.friesian.com/aristotl.htm 1615, http://plato.stanford.edu/entries/aristotle-logic/ 1615, and https://en.wikipedia.org/wiki/Syllogism 1615. (Contributed by David A. Wheeler, 24-Aug-2016.) |
⊢ ∀𝑥(𝜑 → 𝜓) & ⊢ ∀𝑥(𝜒 → 𝜑) ⇒ ⊢ ∀𝑥(𝜒 → 𝜓) | ||
Theorem | celarent 2099 | "Celarent", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, and all 𝜒 is 𝜑, therefore no 𝜒 is 𝜓. (In Aristotelian notation, EAE-1: MeP and SaM therefore SeP.) For example, given the "No reptiles have fur" and "All snakes are reptiles", therefore "No snakes have fur". Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.) |
⊢ ∀𝑥(𝜑 → ¬ 𝜓) & ⊢ ∀𝑥(𝜒 → 𝜑) ⇒ ⊢ ∀𝑥(𝜒 → ¬ 𝜓) | ||
Theorem | darii 2100 | "Darii", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and some 𝜒 is 𝜑, therefore some 𝜒 is 𝜓. (In Aristotelian notation, AII-1: MaP and SiM therefore SiP.) For example, given "All rabbits have fur" and "Some pets are rabbits", therefore "Some pets have fur". Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.) |
⊢ ∀𝑥(𝜑 → 𝜓) & ⊢ ∃𝑥(𝜒 ∧ 𝜑) ⇒ ⊢ ∃𝑥(𝜒 ∧ 𝜓) |
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