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Theorem List for Metamath Proof Explorer - 2801-2900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnecon4ad 2801 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 2802 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 2803 Contrapositive law deduction for inequality. (Contributed by NM, 10-Jun-2006.)
(𝜑 → (𝐴 = 𝐵𝐶 = 𝐷))       (𝜑 → (𝐶𝐷𝐴𝐵))

Theoremnecon1d 2804 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 2805 Contrapositive inference for inequality. (Contributed by NM, 28-Dec-2008.)
(𝜑 → (𝐴 = 𝐵𝐶𝐷))       (𝜑 → (𝐶 = 𝐷𝐴𝐵))

Theoremnecon4d 2806 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 2807 Contrapositive inference for inequality. (Contributed by NM, 23-May-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(𝜑𝐴 = 𝐵)       (𝐴𝐵 → ¬ 𝜑)

Theoremnecon3bi 2808 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 2809 Contrapositive inference for inequality. (Contributed by NM, 12-Feb-2007.) (Proof shortened by Wolf Lammen, 22-Nov-2019.)
𝜑𝐴 = 𝐵)       (𝐴𝐵𝜑)

Theoremnecon1bi 2810 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 2811 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 2812 Contrapositive inference for inequality. (Contributed by NM, 1-Apr-2007.)
(𝜑𝐴𝐵)       (𝐴 = 𝐵 → ¬ 𝜑)

Theoremnecon4ai 2813 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 2814 Contrapositive inference for inequality. (Contributed by NM, 9-Aug-2006.) (Proof shortened by Wolf Lammen, 22-Nov-2019.)
(𝐴 = 𝐵𝐶 = 𝐷)       (𝐶𝐷𝐴𝐵)

Theoremnecon1i 2815 Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.)
(𝐴𝐵𝐶 = 𝐷)       (𝐶𝐷𝐴 = 𝐵)

Theoremnecon2i 2816 Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.)
(𝐴 = 𝐵𝐶𝐷)       (𝐶 = 𝐷𝐴𝐵)

Theoremnecon4i 2817 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 2818 Deduction from equality to inequality. (Contributed by NM, 21-Mar-2007.)
(𝜑 → (𝐴 = 𝐵𝜓))       (𝜑 → (𝐴𝐵 ↔ ¬ 𝜓))

Theoremnecon3bbid 2819 Deduction from equality to inequality. (Contributed by NM, 2-Jun-2007.)
(𝜑 → (𝜓𝐴 = 𝐵))       (𝜑 → (¬ 𝜓𝐴𝐵))

Theoremnecon1abid 2820 Contrapositive deduction for inequality. (Contributed by NM, 21-Aug-2007.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)
(𝜑 → (¬ 𝜓𝐴 = 𝐵))       (𝜑 → (𝐴𝐵𝜓))

Theoremnecon1bbid 2821 Contrapositive inference for inequality. (Contributed by NM, 31-Jan-2008.)
(𝜑 → (𝐴𝐵𝜓))       (𝜑 → (¬ 𝜓𝐴 = 𝐵))

Theoremnecon4abid 2822 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 2823 Contrapositive law deduction for inequality. (Contributed by NM, 9-May-2012.)
(𝜑 → (¬ 𝜓𝐴𝐵))       (𝜑 → (𝜓𝐴 = 𝐵))

Theoremnecon2abid 2824 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 2825 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 2826 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 2827 Contrapositive law deduction for inequality. (Contributed by NM, 29-Jun-2007.)
(𝜑 → (𝐴𝐵𝐶𝐷))       (𝜑 → (𝐴 = 𝐵𝐶 = 𝐷))

Theoremnecon3abii 2828 Deduction from equality to inequality. (Contributed by NM, 9-Nov-2007.)
(𝐴 = 𝐵𝜑)       (𝐴𝐵 ↔ ¬ 𝜑)

Theoremnecon3bbii 2829 Deduction from equality to inequality. (Contributed by NM, 13-Apr-2007.)
(𝜑𝐴 = 𝐵)       𝜑𝐴𝐵)

Theoremnecon1abii 2830 Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
𝜑𝐴 = 𝐵)       (𝐴𝐵𝜑)

Theoremnecon1bbii 2831 Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)
(𝐴𝐵𝜑)       𝜑𝐴 = 𝐵)

Theoremnecon2abii 2832 Contrapositive inference for inequality. (Contributed by NM, 2-Mar-2007.)
(𝐴 = 𝐵 ↔ ¬ 𝜑)       (𝜑𝐴𝐵)

Theoremnecon2bbii 2833 Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)
(𝜑𝐴𝐵)       (𝐴 = 𝐵 ↔ ¬ 𝜑)

Theoremnecon3bii 2834 Inference from equality to inequality. (Contributed by NM, 23-Feb-2005.)
(𝐴 = 𝐵𝐶 = 𝐷)       (𝐴𝐵𝐶𝐷)

Theoremnecom 2835 Commutation of inequality. (Contributed by NM, 14-May-1999.)
(𝐴𝐵𝐵𝐴)

Theoremnecomi 2836 Inference from commutative law for inequality. (Contributed by NM, 17-Oct-2012.)
𝐴𝐵       𝐵𝐴

Theoremnecomd 2837 Deduction from commutative law for inequality. (Contributed by NM, 12-Feb-2008.)
(𝜑𝐴𝐵)       (𝜑𝐵𝐴)

Theoremnesym 2838 Characterization of inequality in terms of reversed equality (see bicom 211). (Contributed by BJ, 7-Jul-2018.)
(𝐴𝐵 ↔ ¬ 𝐵 = 𝐴)

Theoremnesymi 2839 Inference associated with nesym 2838. (Contributed by BJ, 7-Jul-2018.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
𝐴𝐵        ¬ 𝐵 = 𝐴

Theoremnesymir 2840 Inference associated with nesym 2838. (Contributed by BJ, 7-Jul-2018.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
¬ 𝐴 = 𝐵       𝐵𝐴

Theoremneeq1d 2841 Deduction for inequality. (Contributed by NM, 25-Oct-1999.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶𝐵𝐶))

Theoremneeq2d 2842 Deduction for inequality. (Contributed by NM, 25-Oct-1999.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴𝐶𝐵))

Theoremneeq12d 2843 Deduction for inequality. (Contributed by NM, 24-Jul-2012.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶𝐵𝐷))

Theoremneeq1 2844 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.) (Proof shortened by Wolf Lammen, 18-Nov-2019.)
(𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))

Theoremneeq2 2845 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.) (Proof shortened by Wolf Lammen, 18-Nov-2019.)
(𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))

Theoremneeq1i 2846 Inference for inequality. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
𝐴 = 𝐵       (𝐴𝐶𝐵𝐶)

Theoremneeq2i 2847 Inference for inequality. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
𝐴 = 𝐵       (𝐶𝐴𝐶𝐵)

Theoremneeq12i 2848 Inference for inequality. (Contributed by NM, 24-Jul-2012.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝐶𝐵𝐷)

Theoremeqnetrd 2849 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)

Theoremeqnetrrd 2850 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐴𝐶)       (𝜑𝐵𝐶)

Theoremneeqtrd 2851 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
(𝜑𝐴𝐵)    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴𝐶)

Theoremeqnetri 2852 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
𝐴 = 𝐵    &   𝐵𝐶       𝐴𝐶

Theoremeqnetrri 2853 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
𝐴 = 𝐵    &   𝐴𝐶       𝐵𝐶

Theoremneeqtri 2854 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
𝐴𝐵    &   𝐵 = 𝐶       𝐴𝐶

Theoremneeqtrri 2855 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
𝐴𝐵    &   𝐶 = 𝐵       𝐴𝐶

Theoremneeqtrrd 2856 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
(𝜑𝐴𝐵)    &   (𝜑𝐶 = 𝐵)       (𝜑𝐴𝐶)

Theoremsyl5eqner 2857 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 2858 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 2859 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 2860 Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)
(𝜑𝐴𝐵)    &   𝐴 = 𝐶    &   𝐵 = 𝐷       (𝜑𝐶𝐷)

Theorem3netr4g 2861 Substitution of equality into both sides of an inequality. (Contributed by NM, 14-Jun-2012.)
(𝜑𝐴𝐵)    &   𝐶 = 𝐴    &   𝐷 = 𝐵       (𝜑𝐶𝐷)

Theoremnebi 2862 Contraposition law for inequality. (Contributed by NM, 28-Dec-2008.)
((𝐴 = 𝐵𝐶 = 𝐷) ↔ (𝐴𝐵𝐶𝐷))

Theorempm13.18 2863 Theorem *13.18 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
((𝐴 = 𝐵𝐴𝐶) → 𝐵𝐶)

Theorempm13.181 2864 Theorem *13.181 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
((𝐴 = 𝐵𝐵𝐶) → 𝐴𝐶)

Theorempm2.61ine 2865 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 2866 A contradiction implies anything. Equality/inequality deduction form. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐴𝐵)       (𝜑𝜓)

Theorempm2.61ne 2867 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 2868 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 2869 Deduction eliminating an inequality in an antecedent. (Contributed by NM, 30-Nov-2011.)
((𝜑𝐴 = 𝐵) → 𝜓)    &   ((𝜑𝐴𝐵) → 𝜓)       (𝜑𝜓)

Theorempm2.61da2ne 2870 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 2871 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 2872 Equality version of pm2.61ii 176. (Contributed by Scott Fenton, 13-Jun-2013.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
((𝐴𝐶𝐵𝐷) → 𝜑)    &   (𝐴 = 𝐶𝜑)    &   (𝐵 = 𝐷𝜑)       𝜑

Theoremneor 2873 Logical OR with an equality. (Contributed by NM, 29-Apr-2007.)
((𝐴 = 𝐵𝜓) ↔ (𝐴𝐵𝜓))

Theoremneanior 2874 A De Morgan's law for inequality. (Contributed by NM, 18-May-2007.)
((𝐴𝐵𝐶𝐷) ↔ ¬ (𝐴 = 𝐵𝐶 = 𝐷))

Theoremne3anior 2875 A De Morgan's law for inequality. (Contributed by NM, 30-Sep-2013.)
((𝐴𝐵𝐶𝐷𝐸𝐹) ↔ ¬ (𝐴 = 𝐵𝐶 = 𝐷𝐸 = 𝐹))

Theoremneorian 2876 A De Morgan's law for inequality. (Contributed by NM, 18-May-2007.)
((𝐴𝐵𝐶𝐷) ↔ ¬ (𝐴 = 𝐵𝐶 = 𝐷))

Theoremnemtbir 2877 An inference from an inequality, related to modus tollens. (Contributed by NM, 13-Apr-2007.)
𝐴𝐵    &   (𝜑𝐴 = 𝐵)        ¬ 𝜑

Theoremnelne1 2878 Two classes are different if they don't contain the same element. (Contributed by NM, 3-Feb-2012.)
((𝐴𝐵 ∧ ¬ 𝐴𝐶) → 𝐵𝐶)

Theoremnelne2 2879 Two classes are different if they don't belong to the same class. (Contributed by NM, 25-Jun-2012.)
((𝐴𝐶 ∧ ¬ 𝐵𝐶) → 𝐴𝐵)

Theoremnelelne 2880 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 2881 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 2882 Bound-variable hypothesis builder for inequality. (Contributed by NM, 10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥𝐴    &   𝑥𝐵       𝑥 𝐴𝐵

Theoremnfned 2883 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 2884 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 2885 Inference associated with df-nel 2783. (Contributed by BJ, 7-Jul-2018.)
𝐴𝐵        ¬ 𝐴𝐵

Theoremnelir 2886 Inference associated with df-nel 2783. (Contributed by BJ, 7-Jul-2018.)
¬ 𝐴𝐵       𝐴𝐵

Theoremneleq12d 2887 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 2888 Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
(𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))

Theoremneleq2 2889 Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
(𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))

Theoremnfnel 2890 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 2891 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 2892 Negation of negated membership, analogous to nne 2786. (Contributed by Alexander van der Vekens, 18-Jan-2018.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
𝐴𝐵𝐴𝐵)

Theoremelnelne1 2893 Two classes are different if they don't contain the same element. (Contributed by AV, 28-Jan-2020.)
((𝐴𝐵𝐴𝐶) → 𝐵𝐶)

Theoremelnelne2 2894 Two classes are different if they don't belong to the same class. (Contributed by AV, 28-Jan-2020.)
((𝐴𝐶𝐵𝐶) → 𝐴𝐵)

Theoremnelcon3d 2895 Contrapositive law deduction for negated membership. (Contributed by AV, 28-Jan-2020.)
(𝜑 → (𝐴𝐵𝐶𝐷))       (𝜑 → (𝐶𝐷𝐴𝐵))

2.1.5  Restricted quantification

Syntaxwral 2896 Extend wff notation to include restricted universal quantification.
wff 𝑥𝐴 𝜑

Syntaxwrex 2897 Extend wff notation to include restricted existential quantification.
wff 𝑥𝐴 𝜑

Syntaxwreu 2898 Extend wff notation to include restricted existential uniqueness.
wff ∃!𝑥𝐴 𝜑

Syntaxwrmo 2899 Extend wff notation to include restricted "at most one."
wff ∃*𝑥𝐴 𝜑

Syntaxcrab 2900 Extend class notation to include the restricted class abstraction (class builder).
class {𝑥𝐴𝜑}

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