Home Metamath Proof ExplorerTheorem List (p. 27 of 450) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-28698) Hilbert Space Explorer (28699-30221) Users' Mathboxes (30222-44913)

Theorem List for Metamath Proof Explorer - 2601-2700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsb6fALT 2601 Alternate version of sb6f 2536. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))    &   𝑦𝜑       (𝜃 ↔ ∀𝑥(𝑥 = 𝑦𝜑))

Theoremsb5fALT 2602 Alternate version of sb5f 2537. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))    &   𝑦𝜑       (𝜃 ↔ ∃𝑥(𝑥 = 𝑦𝜑))

Theoremnfsb4tALT 2603 Alternate version of nfsb4t 2538. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))       (∀𝑥𝑧𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧𝜃))

Theoremnfsb4ALT 2604 Alternate version of nfsb4 2539. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))    &   𝑧𝜑       (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧𝜃)

Theoremsbi1ALT 2605 Alternate version of sbi1 2075. (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))    &   (𝜏 ↔ ((𝑥 = 𝑦𝜓) ∧ ∃𝑥(𝑥 = 𝑦𝜓)))    &   (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑𝜓))))       (𝜂 → (𝜃𝜏))

Theoremsbi2ALT 2606 Alternate version of sbi2 2309. (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))    &   (𝜏 ↔ ((𝑥 = 𝑦𝜓) ∧ ∃𝑥(𝑥 = 𝑦𝜓)))    &   (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑𝜓))))       ((𝜃𝜏) → 𝜂)

TheoremsbimALT 2607 Alternate version of sbim 2310. (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))    &   (𝜏 ↔ ((𝑥 = 𝑦𝜓) ∧ ∃𝑥(𝑥 = 𝑦𝜓)))    &   (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑𝜓))))       (𝜂 ↔ (𝜃𝜏))

TheoremsbrimALT 2608 Alternate version of sbrim 2312. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜏 ↔ ((𝑥 = 𝑦𝜓) ∧ ∃𝑥(𝑥 = 𝑦𝜓)))    &   (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑𝜓))))    &   𝑥𝜑       (𝜂 ↔ (𝜑𝜏))

TheoremsbanALT 2609 Alternate version of sban 2085. (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))    &   (𝜏 ↔ ((𝑥 = 𝑦𝜓) ∧ ∃𝑥(𝑥 = 𝑦𝜓)))    &   (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑𝜓))))       (𝜂 ↔ (𝜃𝜏))

TheoremsbbiALT 2610 Alternate version of sbbi 2316. (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))    &   (𝜏 ↔ ((𝑥 = 𝑦𝜓) ∧ ∃𝑥(𝑥 = 𝑦𝜓)))    &   (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑𝜓))))       (𝜂 ↔ (𝜃𝜏))

TheoremsblbisALT 2611 Alternate version of sblbis 2318. (Contributed by NM, 19-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))    &   (𝜏 ↔ ((𝑥 = 𝑦𝜓) ∧ ∃𝑥(𝑥 = 𝑦𝜓)))    &   (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑𝜓))))    &   (𝜏𝜒)       (𝜂 ↔ (𝜃𝜒))

TheoremsbieALT 2612 Alternate version of sbie 2543. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 13-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝜃𝜓)

TheoremsbiedALT 2613 Alternate version of sbied 2544. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Jun-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜏 ↔ ((𝑥 = 𝑦𝜓) ∧ ∃𝑥(𝑥 = 𝑦𝜓)))    &   𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (𝜏𝜒))

Theoremsbco2ALT 2614 Alternate version of sbco2 2552. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Sep-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))    &   (𝜏 ↔ ((𝑧 = 𝑦 → ((𝑥 = 𝑧𝜑) ∧ ∃𝑥(𝑥 = 𝑧𝜑))) ∧ ∃𝑧(𝑧 = 𝑦 ∧ ((𝑥 = 𝑧𝜑) ∧ ∃𝑥(𝑥 = 𝑧𝜑)))))    &   𝑧𝜑       (𝜏𝜃)

Theoremsb7fALT 2615* Alternate version of sb7f 2567. (Contributed by NM, 26-Jul-2006.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))    &   𝑧𝜑       (𝜃 ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧𝜑)))

Theoremdfsb7ALT 2616* Alternate version of dfsb7 2284. (Contributed by NM, 28-Jan-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))       (𝜃 ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧𝜑)))

1.6  Uniqueness and unique existence

Theoremdfmoeu 2617* An elementary proof of moeu 2667 in disguise, connecting an expression characterizing uniqueness (df-mo 2621) to that of existential uniqueness (eu6 2658). No particular order of definition is required, as one can be derived from the other. This is shown here and in dfeumo 2618. (Contributed by Wolf Lammen, 27-May-2019.)
((∃𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦)) ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))

Theoremdfeumo 2618* An elementary proof showing the reverse direction of dfmoeu 2617. Here the characterizing expression of existential uniqueness (eu6 2658) is derived from that of uniqueness (df-mo 2621). (Contributed by Wolf Lammen, 3-Oct-2023.)
((∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)) ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))

1.6.1  Uniqueness: the at-most-one quantifier

Syntaxwmo 2619 Extend wff definition to include the at-most-one quantifier ("there exists at most one 𝑥 such that 𝜑").
wff ∃*𝑥𝜑

Theoremmojust 2620* Soundness justification theorem for df-mo 2621 (note that 𝑦 and 𝑧 need not be disjoint, although the weaker theorem with that disjoint variable condition added would be enough to justify the soundness of the definition). (Contributed by NM, 11-Mar-2010.) Added this theorem by adapting the proof of eujust 2655. (Revised by BJ, 30-Sep-2022.)
(∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))

Definitiondf-mo 2621* Define the at-most-one quantifier. The expression ∃*𝑥𝜑 is read "there exists at most one 𝑥 such that 𝜑". This is also called the "uniqueness quantifier" but that expression is also used for the unique existential quantifier df-eu 2653, therefore we avoid that ambiguous name.

Notation of [BellMachover] p. 460, whose definition we show as mo3 2647. For other possible definitions see moeu 2667 and mo4 2649. (Contributed by Wolf Lammen, 27-May-2019.) Make this the definition (which used to be moeu 2667, while this definition was then proved as dfmo 2681). (Revised by BJ, 30-Sep-2022.)

(∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))

Theoremnexmo 2622 Nonexistence implies uniqueness. (Contributed by BJ, 30-Sep-2022.) Avoid ax-11 2160. (Revised by Wolf Lammen, 16-Oct-2022.)
(¬ ∃𝑥𝜑 → ∃*𝑥𝜑)

Theoremexmo 2623 Any proposition holds for some 𝑥 or holds for at most one 𝑥. (Contributed by NM, 8-Mar-1995.) Shorten proof and avoid df-eu 2653. (Revised by BJ, 14-Oct-2022.)
(∃𝑥𝜑 ∨ ∃*𝑥𝜑)

Theoremmoabs 2624 Absorption of existence condition by uniqueness. (Contributed by NM, 4-Nov-2002.) Shorten proof and avoid df-eu 2653. (Revised by BJ, 14-Oct-2022.)
(∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))

Theoremmoim 2625 The at-most-one quantifier reverses implication. (Contributed by NM, 22-Apr-1995.)
(∀𝑥(𝜑𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑))

Theoremmoimi 2626 The at-most-one quantifier reverses implication. (Contributed by NM, 15-Feb-2006.) Remove use of ax-5 1910. (Revised by Steven Nguyen, 9-May-2023.)
(𝜑𝜓)       (∃*𝑥𝜓 → ∃*𝑥𝜑)

TheoremmoimiOLD 2627 Obsolete version of moimi 2626 as of 9-May-2023. The at-most-one quantifier reverses implication. (Contributed by NM, 15-Feb-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)       (∃*𝑥𝜓 → ∃*𝑥𝜑)

Theoremmoimdv 2628* The at-most-one quantifier reverses implication, deduction form. (Contributed by Thierry Arnoux, 25-Feb-2017.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃*𝑥𝜒 → ∃*𝑥𝜓))

Theoremmobi 2629 Equivalence theorem for the at-most-one quantifier. (Contributed by BJ, 7-Oct-2022.) (Proof shortened by Wolf Lammen, 18-Feb-2023.)
(∀𝑥(𝜑𝜓) → (∃*𝑥𝜑 ↔ ∃*𝑥𝜓))

Theoremmobii 2630 Formula-building rule for the at-most-one quantifier (inference form). (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 17-Oct-2016.) Avoid ax-5 1910. (Revised by Wolf Lammen, 24-Sep-2023.)
(𝜓𝜒)       (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)

TheoremmobiiOLD 2631 Obsolete version of mobii 2630 as of 24-Sep-2023. (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜓𝜒)       (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)

Theoremmobidv 2632* Formula-building rule for the at-most-one quantifier (deduction form). (Contributed by Mario Carneiro, 7-Oct-2016.) Reduce axiom dependencies and shorten proof. (Revised by BJ, 7-Oct-2022.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))

Theoremmobid 2633 Formula-building rule for the at-most-one quantifier (deduction form). (Contributed by NM, 8-Mar-1995.) Remove dependency on ax-10 2144, ax-11 2160, ax-13 2389. (Revised by BJ, 14-Oct-2022.) (Proof shortened by Wolf Lammen, 18-Feb-2023.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))

Theoremmoa1 2634 If an implication holds for at most one value, then its consequent holds for at most one value. See also ala1 1813 and exa1 1837. (Contributed by NM, 28-Jul-1995.) (Proof shortened by Wolf Lammen, 22-Dec-2018.) (Revised by BJ, 29-Mar-2021.)
(∃*𝑥(𝜑𝜓) → ∃*𝑥𝜓)

Theoremmoan 2635 "At most one" is still the case when a conjunct is added. (Contributed by NM, 22-Apr-1995.)
(∃*𝑥𝜑 → ∃*𝑥(𝜓𝜑))

Theoremmoani 2636 "At most one" is still true when a conjunct is added. (Contributed by NM, 9-Mar-1995.)
∃*𝑥𝜑       ∃*𝑥(𝜓𝜑)

Theoremmoor 2637 "At most one" is still the case when a disjunct is removed. (Contributed by NM, 5-Apr-2004.)
(∃*𝑥(𝜑𝜓) → ∃*𝑥𝜑)

Theoremmooran1 2638 "At most one" imports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
((∃*𝑥𝜑 ∨ ∃*𝑥𝜓) → ∃*𝑥(𝜑𝜓))

Theoremmooran2 2639 "At most one" exports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(∃*𝑥(𝜑𝜓) → (∃*𝑥𝜑 ∧ ∃*𝑥𝜓))

Theoremnfmo1 2640 Bound-variable hypothesis builder for the at-most-one quantifier. (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.) Adapt to new definition. (Revised by BJ, 1-Oct-2022.)
𝑥∃*𝑥𝜑

Theoremnfmod2 2641 Bound-variable hypothesis builder for the at-most-one quantifier. Usage of this theorem is discouraged because it depends on ax-13 2389. See nfmodv 2642 for a version replacing the distinctor with a disjoint variable condition, not requiring ax-13 2389. (Contributed by Mario Carneiro, 14-Nov-2016.) Avoid df-eu 2653. (Revised by BJ, 14-Oct-2022.) (New usage is discouraged.)
𝑦𝜑    &   ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥∃*𝑦𝜓)

Theoremnfmodv 2642* Bound-variable hypothesis builder for the at-most-one quantifier. See nfmod 2644 for a version without disjoint variable conditions but requiring ax-13 2389. (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by BJ, 28-Jan-2023.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥∃*𝑦𝜓)

Theoremnfmov 2643* Bound-variable hypothesis builder for the at-most-one quantifier. See nfmo 2645 for a version without disjoint variable conditions but requiring ax-13 2389. (Contributed by NM, 9-Mar-1995.) (Revised by Wolf Lammen, 2-Oct-2023.)
𝑥𝜑       𝑥∃*𝑦𝜑

Theoremnfmod 2644 Bound-variable hypothesis builder for the at-most-one quantifier. Deduction version of nfmo 2645. Usage of this theorem is discouraged because it depends on ax-13 2389. Use the weaker nfmodv 2642 when possible. (Contributed by Mario Carneiro, 14-Nov-2016.) (New usage is discouraged.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥∃*𝑦𝜓)

Theoremnfmo 2645 Bound-variable hypothesis builder for the at-most-one quantifier. Note that 𝑥 and 𝑦 need not be disjoint. Usage of this theorem is discouraged because it depends on ax-13 2389. Use the weaker nfmov 2643 when possible. (Contributed by NM, 9-Mar-1995.) (New usage is discouraged.)
𝑥𝜑       𝑥∃*𝑦𝜑

Theoremmof 2646* Version of df-mo 2621 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by NM, 8-Mar-1995.) Extract dfmo 2681 from this proof, and prove mof 2646 from it (as of 30-Sep-2022, directly from df-mo 2621). (Revised by Wolf Lammen, 28-May-2019.) Avoid ax-13 2389. (Revised by Wolf Lammen, 16-Oct-2022.)
𝑦𝜑       (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))

Theoremmo3 2647* Alternate definition of the at-most-one quantifier. 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.) (Proof shortened by Wolf Lammen, 18-Aug-2019.) Remove dependency on ax-13 2389. (Revised by BJ and WL, 29-Jan-2023.)
𝑦𝜑       (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))

Theoremmo 2648* Equivalent definitions of "there exists at most one". (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 2-Dec-2018.)
𝑦𝜑       (∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))

Theoremmo4 2649* At-most-one quantifier expressed using implicit substitution. This theorem is also a direct consequence of mo4f 2650, but this proof is based on fewer axioms.

By the way, swapping 𝑥, 𝑦 and 𝜑, 𝜓 leads to an expression for ∃*𝑦𝜓, which is equivalent to ∃*𝑥𝜑 (is a proof line), so the right hand side is a rare instance of an expression where swapping the quantifiers can be done without ax-11 2160. (Contributed by NM, 26-Jul-1995.) Reduce axiom usage. (Revised by Wolf Lammen, 18-Oct-2023.)

(𝑥 = 𝑦 → (𝜑𝜓))       (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑𝜓) → 𝑥 = 𝑦))

Theoremmo4f 2650* At-most-one quantifier expressed using implicit substitution. Note that the disjoint variable condition on 𝑦, 𝜑 can be replaced by the nonfreeness hypothesis 𝑦𝜑 with essentially the same proof. (Contributed by NM, 10-Apr-2004.) Remove dependency on ax-13 2389. (Revised by Wolf Lammen, 19-Jan-2023.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑𝜓) → 𝑥 = 𝑦))

Theoremmo4OLD 2651* Obsolete version of mo4 2649 as of 18-Oct-2023. (Contributed by NM, 26-Jul-1995.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑𝜓) → 𝑥 = 𝑦))

1.6.2  Unique existence: the unique existential quantifier

Syntaxweu 2652 Extend wff definition to include the unique existential quantifier ("there exists a unique 𝑥 such that 𝜑").
wff ∃!𝑥𝜑

Definitiondf-eu 2653 Define the existential uniqueness quantifier. This expresses unique existence, or existential uniqueness, which is the conjunction of existence (df-ex 1780) and uniqueness (df-mo 2621). The expression ∃!𝑥𝜑 is read "there exists exactly one 𝑥 such that 𝜑 " or "there exists a unique 𝑥 such that 𝜑". This is also called the "uniqueness quantifier" but that expression is also used for the at-most-one quantifier df-mo 2621, therefore we avoid that ambiguous name.

Definition 10.1 of [BellMachover] p. 97; also Definition *14.02 of [WhiteheadRussell] p. 175. Other possible definitions are given by eu1 2693, eu2 2692, eu3v 2654, and eu6 2658. As for double unique existence, beware that the expression ∃!𝑥∃!𝑦𝜑 means "there exists a unique 𝑥 such that there exists a unique 𝑦 such that 𝜑 " which is a weaker property than "there exists exactly one 𝑥 and one 𝑦 such that 𝜑 " (see 2eu4 2740). (Contributed by NM, 12-Aug-1993.) Make this the definition (which used to be eu6 2658, while this definition was then proved as dfeu 2680). (Revised by BJ, 30-Sep-2022.)

(∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))

Theoremeu3v 2654* An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) Replace a nonfreeness hypothesis with a disjoint variable condition on 𝜑, 𝑦 to reduce axiom usage. (Revised by Wolf Lammen, 29-May-2019.)
(∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))

Theoremeujust 2655* Soundness justification theorem for eu6 2658 when this was the definition of the unique existential quantifier (note that 𝑦 and 𝑧 need not be disjoint, although the weaker theorem with that disjoint variable condition added would be enough to justify the soundness of the definition). See eujustALT 2656 for a proof that provides an example of how it can be achieved through the use of dvelim 2472. (Contributed by NM, 11-Mar-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))

TheoremeujustALT 2656* Alternate proof of eujust 2655 illustrating the use of dvelim 2472. (Contributed by NM, 11-Mar-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))

Theoremeu6lem 2657* Lemma of eu6im 2659. A dissection of an idiom characterizing existential uniqueness. (Contributed by NM, 12-Aug-1993.) This used to be the definition of the unique existential quantifier, while df-eu 2653 was then proved as dfeu 2680. (Revised by BJ, 30-Sep-2022.) (Proof shortened by Wolf Lammen, 3-Jan-2023.) Extract common proof lines. (Revised by Wolf Lammen, 3-Mar-2023.)
(∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ (∃𝑦𝑥(𝑥 = 𝑦𝜑) ∧ ∃𝑧𝑥(𝜑𝑥 = 𝑧)))

Theoremeu6 2658* Alternate definition of the unique existential quantifier df-eu 2653 not using the at-most-one quantifier. (Contributed by NM, 12-Aug-1993.) This used to be the definition of the unique existential quantifier, while df-eu 2653 was then proved as dfeu 2680. (Revised by BJ, 30-Sep-2022.) (Proof shortened by Wolf Lammen, 3-Jan-2023.) Remove use of ax-11 2160. (Revised by SN, 21-Sep-2023.)
(∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))

Theoremeu6im 2659* One direction of eu6 2658 needs fewer axioms. (Contributed by Wolf Lammen, 2-Mar-2023.)
(∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃!𝑥𝜑)

Theoremeuf 2660* Version of eu6 2658 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Wolf Lammen, 30-Oct-2018.) Avoid ax-13 2389. (Revised by Wolf Lammen, 16-Oct-2022.)
𝑦𝜑       (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))

Theoremeuex 2661 Existential uniqueness implies existence. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof shortened by Wolf Lammen, 4-Dec-2018.) (Proof shortened by BJ, 7-Oct-2022.)
(∃!𝑥𝜑 → ∃𝑥𝜑)

Theoremeumo 2662 Existential uniqueness implies uniqueness. (Contributed by NM, 23-Mar-1995.)
(∃!𝑥𝜑 → ∃*𝑥𝜑)

Theoremeumoi 2663 Uniqueness inferred from existential uniqueness. (Contributed by NM, 5-Apr-1995.)
∃!𝑥𝜑       ∃*𝑥𝜑

Theoremexmoeub 2664 Existence implies that uniqueness is equivalent to unique existence. (Contributed by NM, 5-Apr-2004.)
(∃𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃!𝑥𝜑))

Theoremexmoeu 2665 Existence is equivalent to uniqueness implying existential uniqueness. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Wolf Lammen, 5-Dec-2018.) (Proof shortened by BJ, 7-Oct-2022.)
(∃𝑥𝜑 ↔ (∃*𝑥𝜑 → ∃!𝑥𝜑))

Theoremmoeuex 2666 Uniqueness implies that existence is equivalent to unique existence. (Contributed by BJ, 7-Oct-2022.)
(∃*𝑥𝜑 → (∃𝑥𝜑 ↔ ∃!𝑥𝜑))

Theoremmoeu 2667 Uniqueness is equivalent to existence implying unique existence. Alternate definition of the at-most-one quantifier, in terms of the existential quantifier and the unique existential quantifier. (Contributed by NM, 8-Mar-1995.) This used to be the definition of the at-most-one quantifier, while df-mo 2621 was then proved as dfmo 2681. (Revised by BJ, 30-Sep-2022.)
(∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))

Theoremeubi 2668 Equivalence theorem for the unique existential quantifier. Theorem *14.271 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.) Reduce dependencies on axioms. (Revised by BJ, 7-Oct-2022.)
(∀𝑥(𝜑𝜓) → (∃!𝑥𝜑 ↔ ∃!𝑥𝜓))

Theoremeubii 2669 Introduce unique existential quantifier to both sides of an equivalence. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 6-Oct-2016.) Avoid ax-5 1910. (Revised by Wolf Lammen, 27-Sep-2023.)
(𝜑𝜓)       (∃!𝑥𝜑 ↔ ∃!𝑥𝜓)

TheoremeubiiOLD 2670 Obsolete version of eubii 2669 as of 27-Sep-2023. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)       (∃!𝑥𝜑 ↔ ∃!𝑥𝜓)

Theoremeubidv 2671* Formula-building rule for unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.) Reduce axiom dependencies and shorten proof. (Revised by BJ, 7-Oct-2022.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))

Theoremeubid 2672 Formula-building rule for the unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.) (Proof shortened by Wolf Lammen, 19-Feb-2023.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))

Theoremnfeu1 2673 Bound-variable hypothesis builder for uniqueness. See also nfeu1ALT 2674. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥∃!𝑥𝜑

Theoremnfeu1ALT 2674 Alternate proof of nfeu1 2673. This illustrates the systematic way of proving nonfreeness in a defined expression: consider the definiens as a tree whose nodes are its subformulas, and prove by tree-induction nonfreeness of each node, starting from the leaves (generally using nfv 1914 or nf* theorems for previously defined expressions) and up to the root. Here, the definiens is a conjunction of two previously defined expressions, which automatically yields the present proof. (Contributed by BJ, 2-Oct-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥∃!𝑥𝜑

Theoremnfeud2 2675 Bound-variable hypothesis builder for uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2389. Check out nfeudw 2676 for a version that replaces the distinctor with a disjoint variable condition, not requiring ax-13 2389. (Contributed by Mario Carneiro, 14-Nov-2016.) (Proof shortened by Wolf Lammen, 4-Oct-2018.) (Proof shortened by BJ, 14-Oct-2022.) (New usage is discouraged.)
𝑦𝜑    &   ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥∃!𝑦𝜓)

Theoremnfeudw 2676* Bound-variable hypothesis builder for the unique existential quantifier. Deduction version of nfeu 2679. Version of nfeud 2677 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 15-Feb-2013.) (Revised by Gino Giotto, 10-Jan-2024.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥∃!𝑦𝜓)

Theoremnfeud 2677 Bound-variable hypothesis builder for the unique existential quantifier. Deduction version of nfeu 2679. Usage of this theorem is discouraged because it depends on ax-13 2389. Use the weaker nfeudw 2676 when possible. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.) (New usage is discouraged.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥∃!𝑦𝜓)

Theoremnfeuw 2678* Bound-variable hypothesis builder for the unique existential quantifier. Version of nfeu 2679 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 8-Mar-1995.) (Revised by Gino Giotto, 10-Jan-2024.)
𝑥𝜑       𝑥∃!𝑦𝜑

Theoremnfeu 2679 Bound-variable hypothesis builder for the unique existential quantifier. Note that 𝑥 and 𝑦 need not be disjoint. Usage of this theorem is discouraged because it depends on ax-13 2389. Use the weaker nfeuw 2678 when possible. (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.) (New usage is discouraged.)
𝑥𝜑       𝑥∃!𝑦𝜑

Theoremdfeu 2680 Rederive df-eu 2653 from the old definition eu6 2658. (Contributed by NM, 23-Mar-1995.) (Proof shortened by Wolf Lammen, 25-May-2019.) (Proof shortened by BJ, 7-Oct-2022.) (Proof modification is discouraged.) Use df-eu 2653 instead. (New usage is discouraged.)
(∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))

Theoremdfmo 2681* Rederive df-mo 2621 from the old definition moeu 2667. (Contributed by Wolf Lammen, 27-May-2019.) (Proof modification is discouraged.) Use df-mo 2621 instead. (New usage is discouraged.)
(∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))

Theoremeuequ 2682* There exists a unique set equal to a given set. Special case of eueqi 3703 proved using only predicate calculus. The proof needs 𝑦 = 𝑧 be free of 𝑥. This is ensured by having 𝑥 and 𝑦 be distinct. Alternately, a distinctor ¬ ∀𝑥𝑥 = 𝑦 could have been used instead. See eueq 3702 and eueqi 3703 for classes. (Contributed by Stefan Allan, 4-Dec-2008.) (Proof shortened by Wolf Lammen, 8-Sep-2019.) Reduce axiom usage. (Revised by Wolf Lammen, 1-Mar-2023.)
∃!𝑥 𝑥 = 𝑦

Theoremsb8eulem 2683* Lemma. Factor out the common proof skeleton of sb8euv 2684 and sb8eu 2685. Variable substitution in unique existential quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Aug-2019.) Factor out common proof lines. (Revised by Wolf Lammen, 9-Feb-2023.)
𝑦[𝑤 / 𝑥]𝜑       (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)

Theoremsb8euv 2684* Variable substitution in unique existential quantifier. Version of sb8eu 2685 requiring more disjoint variables, but fewer axioms. (Contributed by NM, 7-Aug-1994.) (Revised by Wolf Lammen, 7-Feb-2023.)
𝑦𝜑       (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)

Theoremsb8eu 2685 Variable substitution in unique existential quantifier. Usage of this theorem is discouraged because it depends on ax-13 2389. For a version requiring more disjoint variables, but fewer axioms, see sb8euv 2684. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Aug-2019.) (New usage is discouraged.)
𝑦𝜑       (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)

Theoremsb8mo 2686 Variable substitution for the at-most-one quantifier. Usage of this theorem is discouraged because it depends on ax-13 2389. (Contributed by Alexander van der Vekens, 17-Jun-2017.) (New usage is discouraged.)
𝑦𝜑       (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑)

Theoremcbvmow 2687* Rule used to change bound variables, using implicit substitution. Version of cbvmo 2688 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 9-Mar-1995.) (Revised by Gino Giotto, 10-Jan-2024.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)

Theoremcbvmo 2688 Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2389. Use the weaker cbvmow 2687 when possible. (Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon, 8-Jun-2011.) (Proof shortened by Wolf Lammen, 4-Jan-2023.) (New usage is discouraged.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)

Theoremcbveuw 2689* Version of cbveu 2690 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by Gino Giotto, 10-Jan-2024.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)

Theoremcbveu 2690 Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2389. Use the weaker cbveuw 2689 when possible. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (New usage is discouraged.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)

TheoremcbveuALT 2691 Alternative proof of cbveu 2690. Since df-eu 2653 combines two other quantifiers, one can base this theorem on their associated 'change bounded variable' kind of theorems as well. (Contributed by Wolf Lammen, 5-Jan-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)

Theoremeu2 2692* An alternate way of defining existential uniqueness. Definition 6.10 of [TakeutiZaring] p. 26. (Contributed by NM, 8-Jul-1994.) (Proof shortened by Wolf Lammen, 2-Dec-2018.)
𝑦𝜑       (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))

Theoremeu1 2693* An alternate way to express uniqueness used by some authors. Exercise 2(b) of [Margaris] p. 110. (Contributed by NM, 20-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 29-Oct-2018.) Avoid ax-13 2389. (Revised by Wolf Lammen, 7-Feb-2023.)
𝑦𝜑       (∃!𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)))

Theoremeuor 2694 Introduce a disjunct into a unique existential quantifier. For a version requiring disjoint variables, but fewer axioms, see euorv 2695. (Contributed by NM, 21-Oct-2005.)
𝑥𝜑       ((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑𝜓))

Theoremeuorv 2695* Introduce a disjunct into a unique existential quantifier. Version of euor 2694 requiring disjoint variables, but fewer axioms. (Contributed by NM, 23-Mar-1995.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 14-Jan-2023.)
((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑𝜓))

Theoremeuor2 2696 Introduce or eliminate a disjunct in a unique existential quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)
(¬ ∃𝑥𝜑 → (∃!𝑥(𝜑𝜓) ↔ ∃!𝑥𝜓))

Theoremsbmo 2697* Substitution into an at-most-one quantifier. (Contributed by Jeff Madsen, 2-Sep-2009.)
([𝑦 / 𝑥]∃*𝑧𝜑 ↔ ∃*𝑧[𝑦 / 𝑥]𝜑)

Theoremeu4 2698* Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∀𝑥𝑦((𝜑𝜓) → 𝑥 = 𝑦)))

Theoremeuimmo 2699 Existential uniqueness implies uniqueness through reverse implication. (Contributed by NM, 22-Apr-1995.)
(∀𝑥(𝜑𝜓) → (∃!𝑥𝜓 → ∃*𝑥𝜑))

Theoremeuim 2700 Add 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.) (Proof shortened by Wolf Lammen, 1-Oct-2023.)
((∃𝑥𝜑 ∧ ∀𝑥(𝜑𝜓)) → (∃!𝑥𝜓 → ∃!𝑥𝜑))

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-44913
 Copyright terms: Public domain < Previous  Next >