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Theorem List for Metamath Proof Explorer - 1601-1700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnic-ich 1601 Chained inference. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 ⊼ (𝜓𝜓))    &   (𝜓 ⊼ (𝜒𝜒))       (𝜑 ⊼ (𝜒𝜒))

Theoremnic-idbl 1602 Double the terms. Since doubling is the same as negation, this can be viewed as a contraposition inference. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 ⊼ (𝜓𝜓))       ((𝜓𝜓) ⊼ ((𝜑𝜑) ⊼ (𝜑𝜑)))

Theoremnic-bijust 1603 Biconditional justification from Nicod's axiom. For nic-* definitions, the biconditional connective is not used. Instead, definitions are made based on this form. nic-bi1 1604 and nic-bi2 1605 are used to convert the definitions into usable theorems about one side of the implication. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜏𝜏) ⊼ ((𝜏𝜏) ⊼ (𝜏𝜏)))

Theoremnic-bi1 1604 Inference to extract one side of an implication from a definition. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) ⊼ ((𝜑𝜑) ⊼ (𝜓𝜓)))       (𝜑 ⊼ (𝜓𝜓))

Theoremnic-bi2 1605 Inference to extract the other side of an implication from a 'biconditional' definition. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) ⊼ ((𝜑𝜑) ⊼ (𝜓𝜓)))       (𝜓 ⊼ (𝜑𝜑))

Theoremnic-stdmp 1606 Derive the standard modus ponens from nic-mp 1587. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜑𝜓)       𝜓

Theoremnic-luk1 1607 Proof of luk-1 1571 from nic-ax 1589 and nic-mp 1587 (and definitions nic-dfim 1585 and nic-dfneg 1586). Note that the standard axioms ax-1 6, ax-2 7, and ax-3 8 are proved from the Lukasiewicz axioms by theorems ax1 1582, ax2 1583, and ax3 1584. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))

Theoremnic-luk2 1608 Proof of luk-2 1572 from nic-ax 1589 and nic-mp 1587. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
((¬ 𝜑𝜑) → 𝜑)

Theoremnic-luk3 1609 Proof of luk-3 1573 from nic-ax 1589 and nic-mp 1587. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (¬ 𝜑𝜓))

1.3.6  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom

Theoremlukshef-ax1 1610 This alternative axiom for propositional calculus using the Sheffer Stroke was offered by Lukasiewicz in his Selected Works. It improves on Nicod's axiom by reducing its number of variables by one.

This axiom also uses nic-mp 1587 for its constructions.

Here, the axiom is proved as a substitution instance of nic-ax 1589. (Contributed by Anthony Hart, 31-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

((𝜑 ⊼ (𝜒𝜓)) ⊼ ((𝜃 ⊼ (𝜃𝜃)) ⊼ ((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃)))))

Theoremlukshefth1 1611 Lemma for renicax 1613. (Contributed by NM, 31-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((((𝜏𝜓) ⊼ ((𝜑𝜏) ⊼ (𝜑𝜏))) ⊼ (𝜃 ⊼ (𝜃𝜃))) ⊼ (𝜑 ⊼ (𝜓𝜒)))

Theoremlukshefth2 1612 Lemma for renicax 1613. (Contributed by NM, 31-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜏𝜃) ⊼ ((𝜃𝜏) ⊼ (𝜃𝜏)))

Theoremrenicax 1613 A rederivation of nic-ax 1589 from lukshef-ax1 1610, proving that lukshef-ax1 1610 with nic-mp 1587 can be used as a complete axiomatization of propositional calculus. (Contributed by Anthony Hart, 31-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 ⊼ (𝜒𝜓)) ⊼ ((𝜏 ⊼ (𝜏𝜏)) ⊼ ((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃)))))

1.3.7  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms

Theoremtbw-bijust 1614 Justification for tbw-negdf 1615. (Contributed by Anthony Hart, 15-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) ↔ (((𝜑𝜓) → ((𝜓𝜑) → ⊥)) → ⊥))

Theoremtbw-negdf 1615 The definition of negation, in terms of and . (Contributed by Anthony Hart, 15-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((¬ 𝜑 → (𝜑 → ⊥)) → (((𝜑 → ⊥) → ¬ 𝜑) → ⊥)) → ⊥)

Theoremtbw-ax1 1616 The first of four axioms in the Tarski-Bernays-Wajsberg system. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))

Theoremtbw-ax2 1617 The second of four axioms in the Tarski-Bernays-Wajsberg system. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜑))

Theoremtbw-ax3 1618 The third of four axioms in the Tarski-Bernays-Wajsberg system. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → 𝜑) → 𝜑)

Theoremtbw-ax4 1619 The fourth of four axioms in the Tarski-Bernays-Wajsberg system.

This axiom was added to the Tarski-Bernays axiom system (see tb-ax1 31548, tb-ax2 31549, and tb-ax3 31550) by Wajsberg for completeness. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

(⊥ → 𝜑)

Theoremtbwsyl 1620 Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   (𝜓𝜒)       (𝜑𝜒)

Theoremtbwlem1 1621 Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜓𝜒)) → (𝜓 → (𝜑𝜒)))

Theoremtbwlem2 1622 Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜓 → ⊥)) → (((𝜑𝜒) → 𝜃) → (𝜓𝜃)))

Theoremtbwlem3 1623 Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((((𝜑 → ⊥) → 𝜑) → 𝜑) → 𝜓) → 𝜓)

Theoremtbwlem4 1624 Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑 → ⊥) → 𝜓) → ((𝜓 → ⊥) → 𝜑))

Theoremtbwlem5 1625 Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑 → (𝜓 → ⊥)) → ⊥) → 𝜑)

Theoremre1luk1 1626 luk-1 1571 derived from the Tarski-Bernays-Wajsberg axioms. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))

Theoremre1luk2 1627 luk-2 1572 derived from the Tarski-Bernays-Wajsberg axioms. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((¬ 𝜑𝜑) → 𝜑)

Theoremre1luk3 1628 luk-3 1573 derived from the Tarski-Bernays-Wajsberg axioms.

This theorem, along with re1luk1 1626 and re1luk2 1627 proves that tbw-ax1 1616, tbw-ax2 1617, tbw-ax3 1618, and tbw-ax4 1619, with ax-mp 5 can be used as a complete axiom system for all of propositional calculus. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

(𝜑 → (¬ 𝜑𝜓))

1.3.8  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom

Theoremmerco1 1629 A single axiom for propositional calculus offered by Meredith.

This axiom is worthy of note, due to it having only 19 symbols, not counting parentheses. The more well-known meredith 1557 has 21 symbols, sans parentheses.

See merco2 1652 for another axiom of equal length. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

(((((𝜑𝜓) → (𝜒 → ⊥)) → 𝜃) → 𝜏) → ((𝜏𝜑) → (𝜒𝜑)))

Theoremmerco1lem1 1630 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1629. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (⊥ → 𝜒))

Theoremretbwax4 1631 tbw-ax4 1619 rederived from merco1 1629. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(⊥ → 𝜑)

Theoremretbwax2 1632 tbw-ax2 1617 rederived from merco1 1629. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜑))

Theoremmerco1lem2 1633 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1629. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → 𝜒) → (((𝜓𝜏) → (𝜑 → ⊥)) → 𝜒))

Theoremmerco1lem3 1634 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1629. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → (𝜒 → ⊥)) → (𝜒𝜑))

Theoremmerco1lem4 1635 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1629. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → 𝜒) → (𝜓𝜒))

Theoremmerco1lem5 1636 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1629. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((((𝜑 → ⊥) → 𝜒) → 𝜏) → (𝜑𝜏))

Theoremmerco1lem6 1637 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1629. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜑𝜓)) → (𝜒 → (𝜑𝜓)))

Theoremmerco1lem7 1638 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1629. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (((𝜓𝜒) → 𝜓) → 𝜓))

Theoremretbwax3 1639 tbw-ax3 1618 rederived from merco1 1629. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → 𝜑) → 𝜑)

Theoremmerco1lem8 1640 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1629. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ((𝜓 → (𝜓𝜒)) → (𝜓𝜒)))

Theoremmerco1lem9 1641 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1629. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜑𝜓)) → (𝜑𝜓))

Theoremmerco1lem10 1642 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1629. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((((𝜑𝜓) → 𝜒) → (𝜏𝜒)) → 𝜑) → (𝜃𝜑))

Theoremmerco1lem11 1643 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1629. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → (((𝜒 → (𝜑𝜏)) → ⊥) → 𝜓))

Theoremmerco1lem12 1644 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1629. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → (((𝜒 → (𝜑𝜏)) → 𝜑) → 𝜓))

Theoremmerco1lem13 1645 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1629. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((((𝜑𝜓) → (𝜒𝜓)) → 𝜏) → (𝜑𝜏))

Theoremmerco1lem14 1646 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1629. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((((𝜑𝜓) → 𝜓) → 𝜒) → (𝜑𝜒))

Theoremmerco1lem15 1647 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1629. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → (𝜑 → (𝜒𝜓)))

Theoremmerco1lem16 1648 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1629. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑 → (𝜓𝜒)) → 𝜏) → ((𝜑𝜒) → 𝜏))

Theoremmerco1lem17 1649 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1629. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((((𝜑𝜓) → 𝜑) → 𝜒) → 𝜏) → ((𝜑𝜒) → 𝜏))

Theoremmerco1lem18 1650 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1629. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜓𝜒)) → ((𝜓𝜑) → (𝜓𝜒)))

Theoremretbwax1 1651 tbw-ax1 1616 rederived from merco1 1629.

This theorem, along with retbwax2 1632, retbwax3 1639, and retbwax4 1631, shows that merco1 1629 with ax-mp 5 can be used as a complete axiomatization of propositional calculus. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))

1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom

Theoremmerco2 1652 A single axiom for propositional calculus offered by Meredith.

This axiom has 19 symbols, sans auxiliaries. See notes in merco1 1629. (Contributed by Anthony Hart, 7-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

(((𝜑𝜓) → ((⊥ → 𝜒) → 𝜃)) → ((𝜃𝜑) → (𝜏 → (𝜂𝜑))))

Theoremmercolem1 1653 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1652. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → 𝜒) → (𝜓 → (𝜃𝜒)))

Theoremmercolem2 1654 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1652. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → 𝜑) → (𝜒 → (𝜃𝜑)))

Theoremmercolem3 1655 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1652. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜓𝜒) → (𝜓 → (𝜑𝜒)))

Theoremmercolem4 1656 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1652. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜃 → (𝜂𝜑)) → (((𝜃𝜒) → 𝜑) → (𝜏 → (𝜂𝜑))))

Theoremmercolem5 1657 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1652. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 → ((𝜃𝜑) → (𝜏 → (𝜒𝜑))))

Theoremmercolem6 1658 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1652. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒)))

Theoremmercolem7 1659 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1652. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → (((𝜑𝜒) → (𝜃𝜓)) → (𝜃𝜓)))

Theoremmercolem8 1660 Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1652. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜓 → (𝜑𝜒)) → (𝜏 → (𝜃 → (𝜑𝜒)))))

Theoremre1tbw1 1661 tbw-ax1 1616 rederived from merco2 1652. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))

Theoremre1tbw2 1662 tbw-ax2 1617 rederived from merco2 1652. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜑))

Theoremre1tbw3 1663 tbw-ax3 1618 rederived from merco2 1652. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → 𝜑) → 𝜑)

Theoremre1tbw4 1664 tbw-ax4 1619 rederived from merco2 1652.

This theorem, along with re1tbw1 1661, re1tbw2 1662, and re1tbw3 1663, shows that merco2 1652, along with ax-mp 5, can be used as a complete axiomatization of propositional calculus. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

(⊥ → 𝜑)

1.3.10  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms

Theoremrb-bijust 1665 Justification for rb-imdf 1666. (Contributed by Anthony Hart, 17-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) ↔ ¬ (¬ (¬ 𝜑𝜓) ∨ ¬ (¬ 𝜓𝜑)))

Theoremrb-imdf 1666 The definition of implication, in terms of and ¬. (Contributed by Anthony Hart, 17-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ (¬ (¬ (𝜑𝜓) ∨ (¬ 𝜑𝜓)) ∨ ¬ (¬ (¬ 𝜑𝜓) ∨ (𝜑𝜓)))

Theoremanmp 1667 Modus ponens for ¬ axiom systems. (Contributed by Anthony Hart, 12-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   𝜑𝜓)       𝜓

Theoremrb-ax1 1668 The first of four axioms in the Russell-Bernays axiom system. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ (¬ 𝜓𝜒) ∨ (¬ (𝜑𝜓) ∨ (𝜑𝜒)))

Theoremrb-ax2 1669 The second of four axioms in the Russell-Bernays axiom system. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ (𝜑𝜓) ∨ (𝜓𝜑))

Theoremrb-ax3 1670 The third of four axioms in the Russell-Bernays axiom system. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑 ∨ (𝜓𝜑))

Theoremrb-ax4 1671 The fourth of four axioms in the Russell-Bernays axiom system. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ (𝜑𝜑) ∨ 𝜑)

Theoremrbsyl 1672 Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜓𝜒)    &   (𝜑𝜓)       (𝜑𝜒)

Theoremrblem1 1673 Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑𝜓)    &   𝜒𝜃)       (¬ (𝜑𝜒) ∨ (𝜓𝜃))

Theoremrblem2 1674 Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ (𝜒𝜑) ∨ (𝜒 ∨ (𝜑𝜓)))

Theoremrblem3 1675 Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ (𝜒𝜑) ∨ ((𝜒𝜓) ∨ 𝜑))

Theoremrblem4 1676 Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑𝜃)    &   𝜓𝜏)    &   𝜒𝜂)       (¬ ((𝜑𝜓) ∨ 𝜒) ∨ ((𝜂𝜏) ∨ 𝜃))

Theoremrblem5 1677 Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ (¬ ¬ 𝜑𝜓) ∨ (¬ ¬ 𝜓𝜑))

Theoremrblem6 1678 Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ (¬ (¬ 𝜑𝜓) ∨ ¬ (¬ 𝜓𝜑))       𝜑𝜓)

Theoremrblem7 1679 Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ (¬ (¬ 𝜑𝜓) ∨ ¬ (¬ 𝜓𝜑))       𝜓𝜑)

Theoremre1axmp 1680 ax-mp 5 derived from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜑𝜓)       𝜓

Theoremre2luk1 1681 luk-1 1571 derived from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))

Theoremre2luk2 1682 luk-2 1572 derived from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((¬ 𝜑𝜑) → 𝜑)

Theoremre2luk3 1683 luk-3 1573 derived from Russell-Bernays'.

This theorem, along with re1axmp 1680, re2luk1 1681, and re2luk2 1682 shows that rb-ax1 1668, rb-ax2 1669, rb-ax3 1670, and rb-ax4 1671, along with anmp 1667, can be used as a complete axiomatization of propositional calculus. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

(𝜑 → (¬ 𝜑𝜓))

1.3.11  Stoic logic non-modal portion (Chrysippus of Soli)

The Greek Stoics developed a system of logic called Stoic logic. The Stoic Chrysippus, in particular, was often considered one of the greatest logicians of antiquity. Stoic logic is different from Aristotle's system, since it focuses on propositional logic, though later thinkers did combine the systems of the Stoics with Aristotle. Jan Lukasiewicz reports, "For anybody familiar with mathematical logic it is self-evident that the Stoic dialectic is the ancient form of modern propositional logic" ( On the history of the logic of proposition by Jan Lukasiewicz (1934), translated in: Selected Works - Edited by Ludwik Borkowski - Amsterdam, North-Holland, 1970 pp. 197-217, referenced in "History of Logic" https://www.historyoflogic.com/logic-stoics.htm).

In this section we show that the propositional logic system we use (which is non-modal) is at least as strong as the non-modal portion of Stoic logic. We show this by showing that our system assumes or proves all of key features of Stoic logic's non-modal portion (specifically the Stoic logic indemonstrables, themata, and principles).

"In terms of contemporary logic, Stoic syllogistic is best understood as a substructural backwards-working Gentzen-style natural-deduction system that consists of five kinds of axiomatic arguments (the indemonstrables) and four inference rules, called themata. An argument is a syllogism precisely if it either is an indemonstrable or can be reduced to one by means of the themata (Diogenes Laertius (D. L. 7.78))." (Ancient Logic, Stanford Encyclopedia of Philosophy https://plato.stanford.edu/entries/logic-ancient/). There are also a few "principles" that support logical reasoning, discussed below. For more information, see "Stoic Logic" by Susanne Bobzien, especially [Bobzien] p. 110-120, especially for a discussion about the themata (including how they were reconstructed and how they were used). There are differences in the systems we can only partly represent, for example, in Stoic logic "truth and falsehood are temporal properties of assertibles... They can belong to an assertible at one time but not at another" ([Bobzien] p. 87). Stoic logic also included various kinds of modalities, which we do not include here since our basic propositional logic does not include modalities.

A key part of the Stoic logic system is a set of five "indemonstrables" assigned to Chrysippus of Soli by Diogenes Laertius, though in general it is difficult to assign specific ideas to specific thinkers. The indemonstrables are described in, for example, [Lopez-Astorga] p. 11 , [Sanford] p. 39, and [Hitchcock] p. 5. These indemonstrables are modus ponendo ponens (modus ponens) ax-mp 5, modus tollendo tollens (modus tollens) mto 187, modus ponendo tollens I mptnan 1684, modus ponendo tollens II mptxor 1685, and modus tollendo ponens (exclusive-or version) mtpxor 1687. The first is an axiom, the second is already proved; in this section we prove the other three. Note that modus tollendo ponens mtpxor 1687 originally used exclusive-or, but over time the name modus tollendo ponens has increasingly referred to an inclusive-or variation, which is proved in mtpor 1686.

After we prove the indemonstratables, we then prove all the Stoic logic themata (the inference rules of Stoic logic; "thema" is singular). This is straightforward for thema 1 (stoic1a 1688 and stoic1b 1689) and thema 3 (stoic3 1692). However, while Stoic logic was once a leading logic system, most direct information about Stoic logic has since been lost, including the exact texts of thema 2 and thema 4. There are, however, enough references and specific examples to support reconstruction. Themata 2 and 4 have been reconstructed; see statements T2 and T4 in [Bobzien] p. 110-120 and our proofs of them in stoic2a 1690, stoic2b 1691, stoic4a 1693, and stoic4b 1694.

Stoic logic also had a set of principles involving assertibles. Statements in [Bobzien] p. 99 express the known principles. The following paragraphs discuss these principles and our proofs of them.

"A principle of double negation, expressed by saying that a double-negation (Not: not: p) is equivalent to the assertible that is doubly negated (p) (DL VII 69)." In other words, (𝜑 ↔ ¬ ¬ 𝜑) as proven in notnotb 303.

"The principle that all conditionals that are formed by using the same assertible twice (like 'If p, p') are true (Cic. Acad. II 98)." In other words, (𝜑𝜑) as proven in id 22.

"The principle that all disjunctions formed by a contradiction (like 'Either p or not: p') are true (S. E. M VIII 282)" Remember that in Stoic logic, 'or' means 'exclusive or'. In other words, (𝜑 ⊻ ¬ 𝜑) as proven in xorexmid 1472.

[Bobzien] p. 99 also suggests that Stoic logic may have dealt with commutativity (see xorcom 1459 and ancom 465) and the principle of contraposition (con4 111) (pointing to DL VII 194).

In short, the non-modal propositional logic system we use is at least as strong as the non-modal portion of Stoic logic.

For more about Aristotle's system, see barbara 2551 and related theorems.

Theoremmptnan 1684 Modus ponendo tollens 1, one of the "indemonstrables" in Stoic logic. See rule 1 on [Lopez-Astorga] p. 12 , rule 1 on [Sanford] p. 40, and rule A3 in [Hitchcock] p. 5. Sanford describes this rule second (after mptxor 1685) as a "safer, and these days much more common" version of modus ponendo tollens because it avoids confusion between inclusive-or and exclusive-or. (Contributed by David A. Wheeler, 3-Jul-2016.)
𝜑    &    ¬ (𝜑𝜓)        ¬ 𝜓

Theoremmptxor 1685 Modus ponendo tollens 2, one of the "indemonstrables" in Stoic logic. Note that this uses exclusive-or . See rule 2 on [Lopez-Astorga] p. 12 , rule 4 on [Sanford] p. 39 and rule A4 in [Hitchcock] p. 5 . (Contributed by David A. Wheeler, 3-Jul-2016.) (Proof shortened by Wolf Lammen, 12-Nov-2017.) (Proof shortened by BJ, 19-Apr-2019.)
𝜑    &   (𝜑𝜓)        ¬ 𝜓

Theoremmtpor 1686 Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtpxor 1687, one of the five original "indemonstrables" in Stoic logic. However, in Stoic logic this rule used exclusive-or, while the name modus tollendo ponens often refers to a variant of the rule that uses inclusive-or instead. The rule says, "if 𝜑 is not true, and 𝜑 or 𝜓 (or both) are true, then 𝜓 must be true." An alternative phrasing is, "Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth." -- Sherlock Holmes (Sir Arthur Conan Doyle, 1890: The Sign of the Four, ch. 6). (Contributed by David A. Wheeler, 3-Jul-2016.) (Proof shortened by Wolf Lammen, 11-Nov-2017.)
¬ 𝜑    &   (𝜑𝜓)       𝜓

Theoremmtpxor 1687 Modus tollendo ponens (original exclusive-or version), aka disjunctive syllogism, similar to mtpor 1686, one of the five "indemonstrables" in Stoic logic. The rule says, "if 𝜑 is not true, and either 𝜑 or 𝜓 (exclusively) are true, then 𝜓 must be true." Today the name "modus tollendo ponens" often refers to a variant, the inclusive-or version as defined in mtpor 1686. See rule 3 on [Lopez-Astorga] p. 12 (note that the "or" is the same as mptxor 1685, that is, it is exclusive-or df-xor 1457), rule 3 of [Sanford] p. 39 (where it is not as clearly stated which kind of "or" is used but it appears to be in the same sense as mptxor 1685), and rule A5 in [Hitchcock] p. 5 (exclusive-or is expressly used). (Contributed by David A. Wheeler, 4-Jul-2016.) (Proof shortened by Wolf Lammen, 11-Nov-2017.) (Proof shortened by BJ, 19-Apr-2019.)
¬ 𝜑    &   (𝜑𝜓)       𝜓

Theoremstoic1a 1688 Stoic logic Thema 1 (part a).

The first thema of the four Stoic logic themata, in its basic form, was:

"When from two (assertibles) a third follows, then from either of them together with the contradictory of the conclusion the contradictory of the other follows." (Apuleius Int. 209.9-14), see [Bobzien] p. 117 and https://plato.stanford.edu/entries/logic-ancient/

We will represent thema 1 as two very similar rules stoic1a 1688 and stoic1b 1689 to represent each side. (Contributed by David A. Wheeler, 16-Feb-2019.) (Proof shortened by Wolf Lammen, 21-May-2020.)

((𝜑𝜓) → 𝜃)       ((𝜑 ∧ ¬ 𝜃) → ¬ 𝜓)

Theoremstoic1b 1689 Stoic logic Thema 1 (part b). The other part of thema 1 of Stoic logic; see stoic1a 1688. (Contributed by David A. Wheeler, 16-Feb-2019.)
((𝜑𝜓) → 𝜃)       ((𝜓 ∧ ¬ 𝜃) → ¬ 𝜑)

Theoremstoic2a 1690 Stoic logic Thema 2 version a. Statement T2 of [Bobzien] p. 117 shows a reconstructed version of Stoic logic thema 2 as follows: "When from two assertibles a third follows, and from the third and one (or both) of the two another follows, then this other follows from the first two." Bobzien uses constructs such as 𝜑, 𝜓𝜒; in Metamath we will represent that construct as 𝜑𝜓𝜒. This version a is without the phrase "or both"; see stoic2b 1691 for the version with the phrase "or both". We already have this rule as syldan 486, so here we show the equivalence and discourage its use. (New usage is discouraged.) (Contributed by David A. Wheeler, 17-Feb-2019.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜒) → 𝜃)       ((𝜑𝜓) → 𝜃)

Theoremstoic2b 1691 Stoic logic Thema 2 version b. See stoic2a 1690. Version b is with the phrase "or both". We already have this rule as mpd3an3 1417, so here we prove the equivalence and discourage its use. (New usage is discouraged.) (Contributed by David A. Wheeler, 17-Feb-2019.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜓𝜒) → 𝜃)       ((𝜑𝜓) → 𝜃)

Theoremstoic3 1692 Stoic logic Thema 3. Statement T3 of [Bobzien] p. 116-117 discusses Stoic logic Thema 3. "When from two (assemblies) a third follows, and from the one that follows (i.e., the third) together with another, external assumption, another follows, then other follows from the first two and the externally co-assumed one. (Simp. Cael. 237.2-4)" (Contributed by David A. Wheeler, 17-Feb-2019.)
((𝜑𝜓) → 𝜒)    &   ((𝜒𝜃) → 𝜏)       ((𝜑𝜓𝜃) → 𝜏)

Theoremstoic4a 1693 Stoic logic Thema 4 version a. Statement T4 of [Bobzien] p. 117 shows a reconstructed version of Stoic logic Thema 4: "When from two assertibles a third follows, and from the third and one (or both) of the two and one (or more) external assertible(s) another follows, then this other follows from the first two and the external(s)."

We use 𝜃 to represent the "external" assertibles. This is version a, which is without the phrase "or both"; see stoic4b 1694 for the version with the phrase "or both". (Contributed by David A. Wheeler, 17-Feb-2019.)

((𝜑𝜓) → 𝜒)    &   ((𝜒𝜑𝜃) → 𝜏)       ((𝜑𝜓𝜃) → 𝜏)

Theoremstoic4b 1694 Stoic logic Thema 4 version b. This is version b, which is with the phrase "or both". See stoic4a 1693 for more information. (Contributed by David A. Wheeler, 17-Feb-2019.)
((𝜑𝜓) → 𝜒)    &   (((𝜒𝜑𝜓) ∧ 𝜃) → 𝜏)       ((𝜑𝜓𝜃) → 𝜏)

1.4  Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes)

Here we extend the language of wffs with predicate calculus, which allows us to talk about individual objects in a domain of discussion (which for us will be the universe of all sets, so we call them "setvar variables") and make true/false statements about predicates, which are relationships between objects, such as whether or not two objects are equal. In addition, we introduce universal quantification ("for all", e.g. ax-4 1728) in order to make statements about whether a wff holds for every object in the domain of discussion. Later we introduce existential quantification ("there exists", df-ex 1696) which is defined in terms of universal quantification.

Our axioms are really axiom schemes, and our wff and setvar variables are metavariables ranging over expressions in an underlying "object language." This is explained here: mmset.html#axiomnote.

Our axiom system starts with the predicate calculus axiom schemes system S2 of Tarski defined in his 1965 paper, "A Simplified Formalization of Predicate Logic with Identity" [Tarski]. System S2 is defined in the last paragraph on p. 77, and repeated on p. 81 of [KalishMontague]. We do not include scheme B5 (our sp 2041) of system S2 since [KalishMontague] shows it to be logically redundant (Lemma 9, p. 87, which we prove as theorem spw 1954 below).

Theorem spw 1954 can be used to prove any instance of sp 2041 having mutually distinct setvar variables and no wff metavariables. However, it seems that sp 2041 in its general form cannot be derived from only Tarski's schemes. We do not include B5 i.e. sp 2041 as part of what we call "Tarski's system" because we want it to be the smallest set of axioms that is logically complete with no redundancies. We later prove sp 2041 as theorem axc5 33196 using the auxiliary axiom schemes that make our system metalogically complete.

Our version of Tarski's system S2 consists of propositional calculus (ax-mp 5, ax-1 6, ax-2 7, ax-3 8) plus ax-gen 1713, ax-4 1728, ax-5 1827, ax-6 1875, ax-7 1922, ax-8 1979, and ax-9 1986. The last three are equality axioms that represent three sub-schemes of Tarski's scheme B8. Due to its side-condition ("where 𝜑 is an atomic formula and 𝜓 is obtained by replacing an occurrence of the variable 𝑥 by the variable 𝑦"), we cannot represent his B8 directly without greatly complicating our scheme language, but the simpler schemes ax-7 1922, ax-8 1979, and ax-9 1986 are sufficient for set theory and much easier to work with.

Tarski's system is exactly equivalent to the traditional axiom system in most logic textbooks but has the advantage of being easy to manipulate with a computer program, and its simpler metalogic (with no built-in notions of "free variable" and "proper substitution") is arguably easier for a non-logician human to follow step by step in a proof (where "follow" means being able to identify the substitutions that were made, without necessarily a higher-level understanding). In particular, it is logically complete in that it can derive all possible object-language theorems of predicate calculus with equality, i.e. the same theorems as the traditional system can derive.

However, for efficiency (and indeed a key feature that makes Metamath successful), our system is designed to derive reusable theorem schemes (rather than object-language theorems) from other schemes. From this "metalogical" point of view, Tarski's S2 is not complete. For example, we cannot derive scheme sp 2041, even though (using spw 1954) we can derive all instances of it that don't involve wff metavariables or bundled setvar variables. (Two setvar variables are "bundled" if they can be substituted with the same setvar variable i.e. do not have a \$d distinct variable proviso.) Later we will introduce auxiliary axiom schemes ax-10 2006, ax-11 2021, ax-12 2034, and ax-13 2234 that are metatheorems of Tarski's system (i.e. are logically redundant) but which give our system the property of "scheme completeness," allowing us to prove directly (instead of, say, by induction on formula length) all possible schemes that can be expressed in our language.

1.4.1  Universal quantifier (continued); define "exists" and "not free"

The universal quantifier was introduced above in wal 1473 for use by df-tru 1478. See the comments in that section. In this section, we continue with the first "real" use of it.

Syntaxwex 1695 Extend wff definition to include the existential quantifier ("there exists").
wff 𝑥𝜑

Definitiondf-ex 1696 Define existential quantification. 𝑥𝜑 means "there exists at least one set 𝑥 such that 𝜑 is true." Definition of [Margaris] p. 49. (Contributed by NM, 10-Jan-1993.)
(∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)

Theoremalnex 1697 Theorem 19.7 of [Margaris] p. 89. (Contributed by NM, 12-Mar-1993.)
(∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)

Theoremeximal 1698 A utility theorem. An interesting case is when the same formula is substituted for both 𝜑 and 𝜓, since then both implications express a type of non-freeness. See also alimex 1748. (Contributed by BJ, 12-May-2019.)
((∃𝑥𝜑𝜓) ↔ (¬ 𝜓 → ∀𝑥 ¬ 𝜑))

Syntaxwnf 1699 Extend wff definition to include the not-free predicate.
wff 𝑥𝜑

SyntaxwnfOLD 1700 Extend wff definition to include the not-free predicate. (New usage is discouraged.)
wff 𝑥𝜑

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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 42301-42360
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