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

2.1.4  Negated equality and membership

Syntaxwne 2601 Extend wff notation to include inequality.

Syntaxwnel 2602 Extend wff notation to include negated membership.

Definitiondf-ne 2603 Define inequality. (Contributed by NM, 5-Aug-1993.)

Definitiondf-nel 2604 Define negated membership. (Contributed by NM, 7-Aug-1994.)

2.1.4.1  Negated equality

Theoremneii 2605 Inference associated with df-ne 2603. (Contributed by BJ, 7-Jul-2018.)

Theoremneir 2606 Inference associated with df-ne 2603. (Contributed by BJ, 7-Jul-2018.)

Theoremnne 2607 Negation of inequality. (Contributed by NM, 9-Jun-2006.)

Theoremneirr 2608 No class is unequal to itself. Inequality is irreflexive. (Contributed by Stefan O'Rear, 1-Jan-2015.)

Theoremexmidne 2609 Excluded middle with equality and inequality. (Contributed by NM, 3-Feb-2012.)

Theoremnonconne 2610 Law of noncontradiction with equality and inequality. (Contributed by NM, 3-Feb-2012.)

Theoremneeq1 2611 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)

Theoremneeq2 2612 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)

Theoremneeq1i 2613 Inference for inequality. (Contributed by NM, 29-Apr-2005.)

Theoremneeq2i 2614 Inference for inequality. (Contributed by NM, 29-Apr-2005.)

Theoremneeq12i 2615 Inference for inequality. (Contributed by NM, 24-Jul-2012.)

Theoremneeq1d 2616 Deduction for inequality. (Contributed by NM, 25-Oct-1999.)

Theoremneeq2d 2617 Deduction for inequality. (Contributed by NM, 25-Oct-1999.)

Theoremneeq12d 2618 Deduction for inequality. (Contributed by NM, 24-Jul-2012.)

Theoremneneqd 2619 Deduction eliminating inequality definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremeqnetri 2620 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Theoremeqnetrd 2621 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Theoremeqnetrri 2622 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Theoremeqnetrrd 2623 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Theoremneeqtri 2624 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Theoremneeqtrd 2625 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Theoremneeqtrri 2626 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Theoremneeqtrrd 2627 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Theoremsyl5eqner 2628 B chained equality inference for inequality. (Contributed by NM, 6-Jun-2012.)

Theorem3netr3d 2629 Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)

Theorem3netr4d 2630 Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)

Theorem3netr3g 2631 Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)

Theorem3netr4g 2632 Substitution of equality into both sides of an inequality. (Contributed by NM, 14-Jun-2012.)

Theoremnecon3abii 2633 Deduction from equality to inequality. (Contributed by NM, 9-Nov-2007.)

Theoremnecon3bbii 2634 Deduction from equality to inequality. (Contributed by NM, 13-Apr-2007.)

Theoremnecon3bii 2635 Inference from equality to inequality. (Contributed by NM, 23-Feb-2005.)

Theoremnecon3abid 2636 Deduction from equality to inequality. (Contributed by NM, 21-Mar-2007.)

Theoremnecon3bbid 2637 Deduction from equality to inequality. (Contributed by NM, 2-Jun-2007.)

Theoremnecon3bid 2638 Deduction from equality to inequality. (Contributed by NM, 23-Feb-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon3ad 2639 Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon3bd 2640 Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon3d 2641 Contrapositive law deduction for inequality. (Contributed by NM, 10-Jun-2006.)

Theoremnesym 2642 Characterization of inequality in terms of reversed equality (see bicom 193). (Contributed by BJ, 7-Jul-2018.)

Theoremnesymi 2643 Inference associated with nesym 2642. (Contributed by BJ, 7-Jul-2018.)

Theoremnesymir 2644 Inference associated with nesym 2642. (Contributed by BJ, 7-Jul-2018.)

Theoremnecon3i 2645 Contrapositive inference for inequality. (Contributed by NM, 9-Aug-2006.)

Theoremnecon3ai 2646 Contrapositive inference for inequality. (Contributed by NM, 23-May-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon3bi 2647 Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon1ai 2648 Contrapositive inference for inequality. (Contributed by NM, 12-Feb-2007.)

Theoremnecon1bi 2649 Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon1i 2650 Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.)

Theoremnecon2ai 2651 Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon2bi 2652 Contrapositive inference for inequality. (Contributed by NM, 1-Apr-2007.)

Theoremnecon2i 2653 Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.)

Theoremnecon2ad 2654 Contrapositive inference for inequality. (Contributed by NM, 19-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon2bd 2655 Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)

Theoremnecon2d 2656 Contrapositive inference for inequality. (Contributed by NM, 28-Dec-2008.)

Theoremnecon1abii 2657 Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.)

Theoremnecon1bbii 2658 Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.)

Theoremnecon1abid 2659 Contrapositive deduction for inequality. (Contributed by NM, 21-Aug-2007.)

Theoremnecon1bbid 2660 Contrapositive inference for inequality. (Contributed by NM, 31-Jan-2008.)

Theoremnecon2abii 2661 Contrapositive inference for inequality. (Contributed by NM, 2-Mar-2007.)

Theoremnecon2bbii 2662 Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)

Theoremnecon2abid 2663 Contrapositive deduction for inequality. (Contributed by NM, 18-Jul-2007.)

Theoremnecon2bbid 2664 Contrapositive deduction for inequality. (Contributed by NM, 13-Apr-2007.)

Theoremnecon4ai 2665 Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon4i 2666 Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon4ad 2667 Contrapositive inference for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon4bd 2668 Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon4d 2669 Contrapositive inference for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon4abid 2670 Contrapositive law deduction for inequality. (Contributed by NM, 11-Jan-2008.)

Theoremnecon4bbid 2671 Contrapositive law deduction for inequality. (Contributed by NM, 9-May-2012.)

Theoremnecon4bid 2672 Contrapositive law deduction for inequality. (Contributed by NM, 29-Jun-2007.)

Theoremnecon1ad 2673 Contrapositive deduction for inequality. (Contributed by NM, 2-Apr-2007.)

Theoremnecon1bd 2674 Contrapositive deduction for inequality. (Contributed by NM, 21-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon1d 2675 Contrapositive law deduction for inequality. (Contributed by NM, 28-Dec-2008.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremneneqad 2676 If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2619. One-way deduction form of df-ne 2603. (Contributed by David Moews, 28-Feb-2017.)

Theoremnebi 2677 Contraposition law for inequality. (Contributed by NM, 28-Dec-2008.)

Theorempm13.18 2678 Theorem *13.18 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)

Theorempm13.181 2679 Theorem *13.181 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)

Theorempm2.21ddne 2680 A contradiction implies anything. Equality/inequality deduction form. (Contributed by David Moews, 28-Feb-2017.)

Theorempm2.61ne 2681 Deduction eliminating an inequality in an antecedent. (Contributed by NM, 24-May-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theorempm2.61ine 2682 Inference eliminating an inequality in an antecedent. (Contributed by NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theorempm2.61dne 2683 Deduction eliminating an inequality in an antecedent. (Contributed by NM, 1-Jun-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theorempm2.61dane 2684 Deduction eliminating an inequality in an antecedent. (Contributed by NM, 30-Nov-2011.)

Theorempm2.61da2ne 2685 Deduction eliminating two inequalities in an antecedent. (Contributed by NM, 29-May-2013.)

Theorempm2.61da3ne 2686 Deduction eliminating three inequalities in an antecedent. (Contributed by NM, 15-Jun-2013.)

Theoremnecom 2687 Commutation of inequality. (Contributed by NM, 14-May-1999.)

Theoremnecomi 2688 Inference from commutative law for inequality. (Contributed by NM, 17-Oct-2012.)

Theoremnecomd 2689 Deduction from commutative law for inequality. (Contributed by NM, 12-Feb-2008.)

Theoremneor 2690 Logical OR with an equality. (Contributed by NM, 29-Apr-2007.)

Theoremneanior 2691 A De Morgan's law for inequality. (Contributed by NM, 18-May-2007.)

Theoremne3anior 2692 A De Morgan's law for inequality. (Contributed by NM, 30-Sep-2013.)

Theoremneorian 2693 A De Morgan's law for inequality. (Contributed by NM, 18-May-2007.)

Theoremnemtbir 2694 An inference from an inequality, related to modus tollens. (Contributed by NM, 13-Apr-2007.)

Theoremnelne1 2695 Two classes are different if they don't contain the same element. (Contributed by NM, 3-Feb-2012.)

Theoremnelne2 2696 Two classes are different if they don't belong to the same class. (Contributed by NM, 25-Jun-2012.)

Theoremnfne 2697 Bound-variable hypothesis builder for inequality. (Contributed by NM, 10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremnfned 2698 Bound-variable hypothesis builder for inequality. (Contributed by NM, 10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.)

2.1.4.2  Negated membership

Theoremneli 2699 Inference associated with df-nel 2604. (Contributed by BJ, 7-Jul-2018.)

Theoremnelir 2700 Inference associated with df-nel 2604. (Contributed by BJ, 7-Jul-2018.)

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