HomeHome Metamath Proof Explorer
Theorem List (p. 28 of 425)
< 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  Metamath Proof Explorer
(1-26947)
  Hilbert Space Explorer  Hilbert Space Explorer
(26948-28472)
  Users' Mathboxes  Users' Mathboxes
(28473-42426)
 

Theorem List for Metamath Proof Explorer - 2701-2800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnecon2ad 2701 Contrapositive inference for inequality. (Contributed by NM, 19-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 23-Nov-2019.)
(𝜑 → (𝐴 = 𝐵 → ¬ 𝜓))       (𝜑 → (𝜓𝐴𝐵))
 
Theoremnecon2bd 2702 Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)
(𝜑 → (𝜓𝐴𝐵))       (𝜑 → (𝐴 = 𝐵 → ¬ 𝜓))
 
Theoremnecon1ad 2703 Contrapositive deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Wolf Lammen, 23-Nov-2019.)
(𝜑 → (¬ 𝜓𝐴 = 𝐵))       (𝜑 → (𝐴𝐵𝜓))
 
Theoremnecon1bd 2704 Contrapositive deduction for inequality. (Contributed by NM, 21-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 23-Nov-2019.)
(𝜑 → (𝐴𝐵𝜓))       (𝜑 → (¬ 𝜓𝐴 = 𝐵))
 
Theoremnecon4ad 2705 Contrapositive inference for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 23-Nov-2019.)
(𝜑 → (𝐴𝐵 → ¬ 𝜓))       (𝜑 → (𝜓𝐴 = 𝐵))
 
Theoremnecon4bd 2706 Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 23-Nov-2019.)
(𝜑 → (¬ 𝜓𝐴𝐵))       (𝜑 → (𝐴 = 𝐵𝜓))
 
Theoremnecon3d 2707 Contrapositive law deduction for inequality. (Contributed by NM, 10-Jun-2006.)
(𝜑 → (𝐴 = 𝐵𝐶 = 𝐷))       (𝜑 → (𝐶𝐷𝐴𝐵))
 
Theoremnecon1d 2708 Contrapositive law deduction for inequality. (Contributed by NM, 28-Dec-2008.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(𝜑 → (𝐴𝐵𝐶 = 𝐷))       (𝜑 → (𝐶𝐷𝐴 = 𝐵))
 
Theoremnecon2d 2709 Contrapositive inference for inequality. (Contributed by NM, 28-Dec-2008.)
(𝜑 → (𝐴 = 𝐵𝐶𝐷))       (𝜑 → (𝐶 = 𝐷𝐴𝐵))
 
Theoremnecon4d 2710 Contrapositive inference for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(𝜑 → (𝐴𝐵𝐶𝐷))       (𝜑 → (𝐶 = 𝐷𝐴 = 𝐵))
 
Theoremnecon3ai 2711 Contrapositive inference for inequality. (Contributed by NM, 23-May-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(𝜑𝐴 = 𝐵)       (𝐴𝐵 → ¬ 𝜑)
 
Theoremnecon3bi 2712 Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 22-Nov-2019.)
(𝐴 = 𝐵𝜑)       𝜑𝐴𝐵)
 
Theoremnecon1ai 2713 Contrapositive inference for inequality. (Contributed by NM, 12-Feb-2007.) (Proof shortened by Wolf Lammen, 22-Nov-2019.)
𝜑𝐴 = 𝐵)       (𝐴𝐵𝜑)
 
Theoremnecon1bi 2714 Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 22-Nov-2019.)
(𝐴𝐵𝜑)       𝜑𝐴 = 𝐵)
 
Theoremnecon2ai 2715 Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 22-Nov-2019.)
(𝐴 = 𝐵 → ¬ 𝜑)       (𝜑𝐴𝐵)
 
Theoremnecon2bi 2716 Contrapositive inference for inequality. (Contributed by NM, 1-Apr-2007.)
(𝜑𝐴𝐵)       (𝐴 = 𝐵 → ¬ 𝜑)
 
Theoremnecon4ai 2717 Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 22-Nov-2019.)
(𝐴𝐵 → ¬ 𝜑)       (𝜑𝐴 = 𝐵)
 
Theoremnecon3i 2718 Contrapositive inference for inequality. (Contributed by NM, 9-Aug-2006.) (Proof shortened by Wolf Lammen, 22-Nov-2019.)
(𝐴 = 𝐵𝐶 = 𝐷)       (𝐶𝐷𝐴𝐵)
 
Theoremnecon1i 2719 Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.)
(𝐴𝐵𝐶 = 𝐷)       (𝐶𝐷𝐴 = 𝐵)
 
Theoremnecon2i 2720 Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.)
(𝐴 = 𝐵𝐶𝐷)       (𝐶 = 𝐷𝐴𝐵)
 
Theoremnecon4i 2721 Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)
(𝐴𝐵𝐶𝐷)       (𝐶 = 𝐷𝐴 = 𝐵)
 
Theoremnecon3abid 2722 Deduction from equality to inequality. (Contributed by NM, 21-Mar-2007.)
(𝜑 → (𝐴 = 𝐵𝜓))       (𝜑 → (𝐴𝐵 ↔ ¬ 𝜓))
 
Theoremnecon3bbid 2723 Deduction from equality to inequality. (Contributed by NM, 2-Jun-2007.)
(𝜑 → (𝜓𝐴 = 𝐵))       (𝜑 → (¬ 𝜓𝐴𝐵))
 
Theoremnecon1abid 2724 Contrapositive deduction for inequality. (Contributed by NM, 21-Aug-2007.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)
(𝜑 → (¬ 𝜓𝐴 = 𝐵))       (𝜑 → (𝐴𝐵𝜓))
 
Theoremnecon1bbid 2725 Contrapositive inference for inequality. (Contributed by NM, 31-Jan-2008.)
(𝜑 → (𝐴𝐵𝜓))       (𝜑 → (¬ 𝜓𝐴 = 𝐵))
 
Theoremnecon4abid 2726 Contrapositive law deduction for inequality. (Contributed by NM, 11-Jan-2008.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)
(𝜑 → (𝐴𝐵 ↔ ¬ 𝜓))       (𝜑 → (𝐴 = 𝐵𝜓))
 
Theoremnecon4bbid 2727 Contrapositive law deduction for inequality. (Contributed by NM, 9-May-2012.)
(𝜑 → (¬ 𝜓𝐴𝐵))       (𝜑 → (𝜓𝐴 = 𝐵))
 
Theoremnecon2abid 2728 Contrapositive deduction for inequality. (Contributed by NM, 18-Jul-2007.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)
(𝜑 → (𝐴 = 𝐵 ↔ ¬ 𝜓))       (𝜑 → (𝜓𝐴𝐵))
 
Theoremnecon2bbid 2729 Contrapositive deduction for inequality. (Contributed by NM, 13-Apr-2007.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)
(𝜑 → (𝜓𝐴𝐵))       (𝜑 → (𝐴 = 𝐵 ↔ ¬ 𝜓))
 
Theoremnecon3bid 2730 Deduction from equality to inequality. (Contributed by NM, 23-Feb-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(𝜑 → (𝐴 = 𝐵𝐶 = 𝐷))       (𝜑 → (𝐴𝐵𝐶𝐷))
 
Theoremnecon4bid 2731 Contrapositive law deduction for inequality. (Contributed by NM, 29-Jun-2007.)
(𝜑 → (𝐴𝐵𝐶𝐷))       (𝜑 → (𝐴 = 𝐵𝐶 = 𝐷))
 
Theoremnecon3abii 2732 Deduction from equality to inequality. (Contributed by NM, 9-Nov-2007.)
(𝐴 = 𝐵𝜑)       (𝐴𝐵 ↔ ¬ 𝜑)
 
Theoremnecon3bbii 2733 Deduction from equality to inequality. (Contributed by NM, 13-Apr-2007.)
(𝜑𝐴 = 𝐵)       𝜑𝐴𝐵)
 
Theoremnecon1abii 2734 Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
𝜑𝐴 = 𝐵)       (𝐴𝐵𝜑)
 
Theoremnecon1bbii 2735 Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)
(𝐴𝐵𝜑)       𝜑𝐴 = 𝐵)
 
Theoremnecon2abii 2736 Contrapositive inference for inequality. (Contributed by NM, 2-Mar-2007.)
(𝐴 = 𝐵 ↔ ¬ 𝜑)       (𝜑𝐴𝐵)
 
Theoremnecon2bbii 2737 Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)
(𝜑𝐴𝐵)       (𝐴 = 𝐵 ↔ ¬ 𝜑)
 
Theoremnecon3bii 2738 Inference from equality to inequality. (Contributed by NM, 23-Feb-2005.)
(𝐴 = 𝐵𝐶 = 𝐷)       (𝐴𝐵𝐶𝐷)
 
Theoremnecom 2739 Commutation of inequality. (Contributed by NM, 14-May-1999.)
(𝐴𝐵𝐵𝐴)
 
Theoremnecomi 2740 Inference from commutative law for inequality. (Contributed by NM, 17-Oct-2012.)
𝐴𝐵       𝐵𝐴
 
Theoremnecomd 2741 Deduction from commutative law for inequality. (Contributed by NM, 12-Feb-2008.)
(𝜑𝐴𝐵)       (𝜑𝐵𝐴)
 
Theoremnesym 2742 Characterization of inequality in terms of reversed equality (see bicom 210). (Contributed by BJ, 7-Jul-2018.)
(𝐴𝐵 ↔ ¬ 𝐵 = 𝐴)
 
Theoremnesymi 2743 Inference associated with nesym 2742. (Contributed by BJ, 7-Jul-2018.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
𝐴𝐵        ¬ 𝐵 = 𝐴
 
Theoremnesymir 2744 Inference associated with nesym 2742. (Contributed by BJ, 7-Jul-2018.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
¬ 𝐴 = 𝐵       𝐵𝐴
 
Theoremneeq1d 2745 Deduction for inequality. (Contributed by NM, 25-Oct-1999.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶𝐵𝐶))
 
Theoremneeq2d 2746 Deduction for inequality. (Contributed by NM, 25-Oct-1999.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴𝐶𝐵))
 
Theoremneeq12d 2747 Deduction for inequality. (Contributed by NM, 24-Jul-2012.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶𝐵𝐷))
 
Theoremneeq1 2748 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.) (Proof shortened by Wolf Lammen, 18-Nov-2019.)
(𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
 
Theoremneeq2 2749 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.) (Proof shortened by Wolf Lammen, 18-Nov-2019.)
(𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))
 
Theoremneeq1i 2750 Inference for inequality. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
𝐴 = 𝐵       (𝐴𝐶𝐵𝐶)
 
Theoremneeq2i 2751 Inference for inequality. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
𝐴 = 𝐵       (𝐶𝐴𝐶𝐵)
 
Theoremneeq12i 2752 Inference for inequality. (Contributed by NM, 24-Jul-2012.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝐶𝐵𝐷)
 
Theoremeqnetrd 2753 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)
 
Theoremeqnetrrd 2754 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐴𝐶)       (𝜑𝐵𝐶)
 
Theoremneeqtrd 2755 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
(𝜑𝐴𝐵)    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴𝐶)
 
Theoremeqnetri 2756 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
𝐴 = 𝐵    &   𝐵𝐶       𝐴𝐶
 
Theoremeqnetrri 2757 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
𝐴 = 𝐵    &   𝐴𝐶       𝐵𝐶
 
Theoremneeqtri 2758 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
𝐴𝐵    &   𝐵 = 𝐶       𝐴𝐶
 
Theoremneeqtrri 2759 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
𝐴𝐵    &   𝐶 = 𝐵       𝐴𝐶
 
Theoremneeqtrrd 2760 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
(𝜑𝐴𝐵)    &   (𝜑𝐶 = 𝐵)       (𝜑𝐴𝐶)
 
Theoremsyl5eqner 2761 A chained equality inference for inequality. (Contributed by NM, 6-Jun-2012.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
𝐵 = 𝐴    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)
 
Theorem3netr3d 2762 Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
(𝜑𝐴𝐵)    &   (𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑𝐶𝐷)
 
Theorem3netr4d 2763 Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.) (Proof shortened by Wolf Lammen, 21-Nov-2019.)
(𝜑𝐴𝐵)    &   (𝜑𝐶 = 𝐴)    &   (𝜑𝐷 = 𝐵)       (𝜑𝐶𝐷)
 
Theorem3netr3g 2764 Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)
(𝜑𝐴𝐵)    &   𝐴 = 𝐶    &   𝐵 = 𝐷       (𝜑𝐶𝐷)
 
Theorem3netr4g 2765 Substitution of equality into both sides of an inequality. (Contributed by NM, 14-Jun-2012.)
(𝜑𝐴𝐵)    &   𝐶 = 𝐴    &   𝐷 = 𝐵       (𝜑𝐶𝐷)
 
Theoremnebi 2766 Contraposition law for inequality. (Contributed by NM, 28-Dec-2008.)
((𝐴 = 𝐵𝐶 = 𝐷) ↔ (𝐴𝐵𝐶𝐷))
 
Theorempm13.18 2767 Theorem *13.18 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
((𝐴 = 𝐵𝐴𝐶) → 𝐵𝐶)
 
Theorempm13.181 2768 Theorem *13.181 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
((𝐴 = 𝐵𝐵𝐶) → 𝐴𝐶)
 
Theorempm2.61ine 2769 Inference eliminating an inequality in an antecedent. (Contributed by NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(𝐴 = 𝐵𝜑)    &   (𝐴𝐵𝜑)       𝜑
 
Theorempm2.21ddne 2770 A contradiction implies anything. Equality/inequality deduction form. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐴𝐵)       (𝜑𝜓)
 
Theorempm2.61ne 2771 Deduction eliminating an inequality in an antecedent. (Contributed by NM, 24-May-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
(𝐴 = 𝐵 → (𝜓𝜒))    &   ((𝜑𝐴𝐵) → 𝜓)    &   (𝜑𝜒)       (𝜑𝜓)
 
Theorempm2.61dne 2772 Deduction eliminating an inequality in an antecedent. (Contributed by NM, 1-Jun-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(𝜑 → (𝐴 = 𝐵𝜓))    &   (𝜑 → (𝐴𝐵𝜓))       (𝜑𝜓)
 
Theorempm2.61dane 2773 Deduction eliminating an inequality in an antecedent. (Contributed by NM, 30-Nov-2011.)
((𝜑𝐴 = 𝐵) → 𝜓)    &   ((𝜑𝐴𝐵) → 𝜓)       (𝜑𝜓)
 
Theorempm2.61da2ne 2774 Deduction eliminating two inequalities in an antecedent. (Contributed by NM, 29-May-2013.)
((𝜑𝐴 = 𝐵) → 𝜓)    &   ((𝜑𝐶 = 𝐷) → 𝜓)    &   ((𝜑 ∧ (𝐴𝐵𝐶𝐷)) → 𝜓)       (𝜑𝜓)
 
Theorempm2.61da3ne 2775 Deduction eliminating three inequalities in an antecedent. (Contributed by NM, 15-Jun-2013.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
((𝜑𝐴 = 𝐵) → 𝜓)    &   ((𝜑𝐶 = 𝐷) → 𝜓)    &   ((𝜑𝐸 = 𝐹) → 𝜓)    &   ((𝜑 ∧ (𝐴𝐵𝐶𝐷𝐸𝐹)) → 𝜓)       (𝜑𝜓)
 
Theorempm2.61iine 2776 Equality version of pm2.61ii 175. (Contributed by Scott Fenton, 13-Jun-2013.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
((𝐴𝐶𝐵𝐷) → 𝜑)    &   (𝐴 = 𝐶𝜑)    &   (𝐵 = 𝐷𝜑)       𝜑
 
Theoremneor 2777 Logical OR with an equality. (Contributed by NM, 29-Apr-2007.)
((𝐴 = 𝐵𝜓) ↔ (𝐴𝐵𝜓))
 
Theoremneanior 2778 A De Morgan's law for inequality. (Contributed by NM, 18-May-2007.)
((𝐴𝐵𝐶𝐷) ↔ ¬ (𝐴 = 𝐵𝐶 = 𝐷))
 
Theoremne3anior 2779 A De Morgan's law for inequality. (Contributed by NM, 30-Sep-2013.)
((𝐴𝐵𝐶𝐷𝐸𝐹) ↔ ¬ (𝐴 = 𝐵𝐶 = 𝐷𝐸 = 𝐹))
 
Theoremneorian 2780 A De Morgan's law for inequality. (Contributed by NM, 18-May-2007.)
((𝐴𝐵𝐶𝐷) ↔ ¬ (𝐴 = 𝐵𝐶 = 𝐷))
 
Theoremnemtbir 2781 An inference from an inequality, related to modus tollens. (Contributed by NM, 13-Apr-2007.)
𝐴𝐵    &   (𝜑𝐴 = 𝐵)        ¬ 𝜑
 
Theoremnelne1 2782 Two classes are different if they don't contain the same element. (Contributed by NM, 3-Feb-2012.)
((𝐴𝐵 ∧ ¬ 𝐴𝐶) → 𝐵𝐶)
 
Theoremnelne2 2783 Two classes are different if they don't belong to the same class. (Contributed by NM, 25-Jun-2012.)
((𝐴𝐶 ∧ ¬ 𝐵𝐶) → 𝐴𝐵)
 
Theoremnelelne 2784 Two classes are different if they don't belong to the same class. (Contributed by Rodolfo Medina, 17-Oct-2010.) (Proof shortened by AV, 10-May-2020.)
𝐴𝐵 → (𝐶𝐵𝐶𝐴))
 
Theoremneneor 2785 If two classes are different, a third class must be different of at least one of them. (Contributed by Thierry Arnoux, 8-Aug-2020.)
(𝐴𝐵 → (𝐴𝐶𝐵𝐶))
 
Theoremnfne 2786 Bound-variable hypothesis builder for inequality. (Contributed by NM, 10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥𝐴    &   𝑥𝐵       𝑥 𝐴𝐵
 
Theoremnfned 2787 Bound-variable hypothesis builder for inequality. (Contributed by NM, 10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑 → Ⅎ𝑥 𝐴𝐵)
 
Theoremnabbi 2788 Not equivalent wff's correspond to not equal class abstractions. (Contributed by AV, 7-Apr-2019.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
(∃𝑥(𝜑 ↔ ¬ 𝜓) ↔ {𝑥𝜑} ≠ {𝑥𝜓})
 
2.1.4.2  Negated membership
 
Theoremneli 2789 Inference associated with df-nel 2687. (Contributed by BJ, 7-Jul-2018.)
𝐴𝐵        ¬ 𝐴𝐵
 
Theoremnelir 2790 Inference associated with df-nel 2687. (Contributed by BJ, 7-Jul-2018.)
¬ 𝐴𝐵       𝐴𝐵
 
Theoremneleq12d 2791 Equality theorem for negated membership. (Contributed by FL, 10-Aug-2016.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶𝐵𝐷))
 
Theoremneleq1 2792 Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
(𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
 
Theoremneleq2 2793 Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
(𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))
 
Theoremnfnel 2794 Bound-variable hypothesis builder for negated membership. (Contributed by David Abernethy, 26-Jun-2011.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥𝐴    &   𝑥𝐵       𝑥 𝐴𝐵
 
Theoremnfneld 2795 Bound-variable hypothesis builder for negated membership. (Contributed by David Abernethy, 26-Jun-2011.) (Revised by Mario Carneiro, 7-Oct-2016.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑 → Ⅎ𝑥 𝐴𝐵)
 
Theoremnnel 2796 Negation of negated membership, analogous to nne 2690. (Contributed by Alexander van der Vekens, 18-Jan-2018.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
𝐴𝐵𝐴𝐵)
 
Theoremelnelne1 2797 Two classes are different if they don't contain the same element. (Contributed by AV, 28-Jan-2020.)
((𝐴𝐵𝐴𝐶) → 𝐵𝐶)
 
Theoremelnelne2 2798 Two classes are different if they don't belong to the same class. (Contributed by AV, 28-Jan-2020.)
((𝐴𝐶𝐵𝐶) → 𝐴𝐵)
 
Theoremnelcon3d 2799 Contrapositive law deduction for negated membership. (Contributed by AV, 28-Jan-2020.)
(𝜑 → (𝐴𝐵𝐶𝐷))       (𝜑 → (𝐶𝐷𝐴𝐵))
 
2.1.5  Restricted quantification
 
Syntaxwral 2800 Extend wff notation to include restricted universal quantification.
wff 𝑥𝐴 𝜑
    < Previous  Next >

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 42401-42426
  Copyright terms: Public domain < Previous  Next >