P. 31 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 3001-3100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	necon4bbid 3001	Contrapositive law deduction for inequality. (Contributed by NM, 9-May-2012.)
⊢ (𝜑 → (¬ 𝜓 ↔ 𝐴 ≠ 𝐵))    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝐴 = 𝐵))
Theorem	necon2abid 3002	Contrapositive deduction for inequality. (Contributed by NM, 18-Jul-2007.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)
⊢ (𝜑 → (𝐴 = 𝐵 ↔ ¬ 𝜓))    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝐴 ≠ 𝐵))
Theorem	necon2bbid 3003	Contrapositive deduction for inequality. (Contributed by NM, 13-Apr-2007.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)
⊢ (𝜑 → (𝜓 ↔ 𝐴 ≠ 𝐵))    ⇒   ⊢ (𝜑 → (𝐴 = 𝐵 ↔ ¬ 𝜓))
Theorem	necon3bid 3004	Deduction from equality to inequality. (Contributed by NM, 23-Feb-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷))    ⇒   ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷))
Theorem	necon4bid 3005	Contrapositive law deduction for inequality. (Contributed by NM, 29-Jun-2007.)
⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷))    ⇒   ⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷))
Theorem	necon3abii 3006	Deduction from equality to inequality. (Contributed by NM, 9-Nov-2007.)
⊢ (𝐴 = 𝐵 ↔ 𝜑)    ⇒   ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝜑)
Theorem	necon3bbii 3007	Deduction from equality to inequality. (Contributed by NM, 13-Apr-2007.)
⊢ (𝜑 ↔ 𝐴 = 𝐵)    ⇒   ⊢ (¬ 𝜑 ↔ 𝐴 ≠ 𝐵)
Theorem	necon1abii 3008	Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
⊢ (¬ 𝜑 ↔ 𝐴 = 𝐵)    ⇒   ⊢ (𝐴 ≠ 𝐵 ↔ 𝜑)
Theorem	necon1bbii 3009	Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)
⊢ (𝐴 ≠ 𝐵 ↔ 𝜑)    ⇒   ⊢ (¬ 𝜑 ↔ 𝐴 = 𝐵)
Theorem	necon2abii 3010	Contrapositive inference for inequality. (Contributed by NM, 2-Mar-2007.)
⊢ (𝐴 = 𝐵 ↔ ¬ 𝜑)    ⇒   ⊢ (𝜑 ↔ 𝐴 ≠ 𝐵)
Theorem	necon2bbii 3011	Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)
⊢ (𝜑 ↔ 𝐴 ≠ 𝐵)    ⇒   ⊢ (𝐴 = 𝐵 ↔ ¬ 𝜑)
Theorem	necon3bii 3012	Inference from equality to inequality. (Contributed by NM, 23-Feb-2005.)
⊢ (𝐴 = 𝐵 ↔ 𝐶 = 𝐷)    ⇒   ⊢ (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷)
Theorem	necom 3013	Commutation of inequality. (Contributed by NM, 14-May-1999.)
⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴)
Theorem	necomi 3014	Inference from commutative law for inequality. (Contributed by NM, 17-Oct-2012.)
⊢ 𝐴 ≠ 𝐵    ⇒   ⊢ 𝐵 ≠ 𝐴
Theorem	necomd 3015	Deduction from commutative law for inequality. (Contributed by NM, 12-Feb-2008.)
⊢ (𝜑 → 𝐴 ≠ 𝐵)    ⇒   ⊢ (𝜑 → 𝐵 ≠ 𝐴)
Theorem	nesym 3016	Characterization of inequality in terms of reversed equality (see bicom 225). (Contributed by BJ, 7-Jul-2018.)
⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐵 = 𝐴)
Theorem	nesymi 3017	Inference associated with nesym 3016. (Contributed by BJ, 7-Jul-2018.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
⊢ 𝐴 ≠ 𝐵    ⇒   ⊢ ¬ 𝐵 = 𝐴
Theorem	nesymir 3018	Inference associated with nesym 3016. (Contributed by BJ, 7-Jul-2018.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
⊢ ¬ 𝐴 = 𝐵    ⇒   ⊢ 𝐵 ≠ 𝐴
Theorem	neeq1d 3019	Deduction for inequality. (Contributed by NM, 25-Oct-1999.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶))
Theorem	neeq2d 3020	Deduction for inequality. (Contributed by NM, 25-Oct-1999.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵))
Theorem	neeq12d 3021	Deduction for inequality. (Contributed by NM, 24-Jul-2012.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐷))
Theorem	neeq1 3022	Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.) (Proof shortened by Wolf Lammen, 18-Nov-2019.)
⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶))
Theorem	neeq2 3023	Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.) (Proof shortened by Wolf Lammen, 18-Nov-2019.)
⊢ (𝐴 = 𝐵 → (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵))
Theorem	neeq1i 3024	Inference for inequality. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶)
Theorem	neeq2i 3025	Inference for inequality. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵)
Theorem	neeq12i 3026	Inference for inequality. (Contributed by NM, 24-Jul-2012.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐷)
Theorem	eqnetrd 3027	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐵 ≠ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐶)
Theorem	eqnetrrd 3028	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐶)    ⇒   ⊢ (𝜑 → 𝐵 ≠ 𝐶)
Theorem	neeqtrd 3029	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐶)
Theorem	eqnetri 3030	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐵 ≠ 𝐶    ⇒   ⊢ 𝐴 ≠ 𝐶
Theorem	eqnetrri 3031	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐴 ≠ 𝐶    ⇒   ⊢ 𝐵 ≠ 𝐶
Theorem	neeqtri 3032	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
⊢ 𝐴 ≠ 𝐵    &   ⊢ 𝐵 = 𝐶    ⇒   ⊢ 𝐴 ≠ 𝐶
Theorem	neeqtrri 3033	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
⊢ 𝐴 ≠ 𝐵    &   ⊢ 𝐶 = 𝐵    ⇒   ⊢ 𝐴 ≠ 𝐶
Theorem	neeqtrrd 3034	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐶)
Theorem	eqnetrrid 3035	A chained equality inference for inequality. (Contributed by NM, 6-Jun-2012.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
⊢ 𝐵 = 𝐴    &   ⊢ (𝜑 → 𝐵 ≠ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐶)
Theorem	3netr3d 3036	Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → 𝐴 = 𝐶)    &   ⊢ (𝜑 → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → 𝐶 ≠ 𝐷)
Theorem	3netr4d 3037	Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.) (Proof shortened by Wolf Lammen, 21-Nov-2019.)
⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐴)    &   ⊢ (𝜑 → 𝐷 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐶 ≠ 𝐷)
Theorem	3netr3g 3038	Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)
⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ 𝐴 = 𝐶    &   ⊢ 𝐵 = 𝐷    ⇒   ⊢ (𝜑 → 𝐶 ≠ 𝐷)
Theorem	3netr4g 3039	Substitution of equality into both sides of an inequality. (Contributed by NM, 14-Jun-2012.)
⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ 𝐶 = 𝐴    &   ⊢ 𝐷 = 𝐵    ⇒   ⊢ (𝜑 → 𝐶 ≠ 𝐷)
Theorem	nebi 3040	Contraposition law for inequality. (Contributed by NM, 28-Dec-2008.)
⊢ ((𝐴 = 𝐵 ↔ 𝐶 = 𝐷) ↔ (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷))
Theorem	pm13.18 3041	Theorem *13.18 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.) (Proof shortened by Wolf Lammen, 29-Oct-2024.)
⊢ ((𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐶) → 𝐵 ≠ 𝐶)
Theorem	pm13.181 3042	Theorem *13.181 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.) (Proof shortened by Wolf Lammen, 30-Oct-2024.)
⊢ ((𝐴 = 𝐵 ∧ 𝐵 ≠ 𝐶) → 𝐴 ≠ 𝐶)
Theorem	pm2.61ine 3043	Inference eliminating an inequality in an antecedent. (Contributed by NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ (𝐴 = 𝐵 → 𝜑)    &   ⊢ (𝐴 ≠ 𝐵 → 𝜑)    ⇒   ⊢ 𝜑
Theorem	pm2.21ddne 3044	A contradiction implies anything. Equality/inequality deduction form. (Contributed by David Moews, 28-Feb-2017.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	pm2.61ne 3045	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 → (𝜓 ↔ 𝜒))    &   ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝜓)    &   ⊢ (𝜑 → 𝜒)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	pm2.61dne 3046	Deduction eliminating an inequality in an antecedent. (Contributed by NM, 1-Jun-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ (𝜑 → (𝐴 = 𝐵 → 𝜓))    &   ⊢ (𝜑 → (𝐴 ≠ 𝐵 → 𝜓))    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	pm2.61dane 3047	Deduction eliminating an inequality in an antecedent. (Contributed by NM, 30-Nov-2011.)
⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝜓)    &   ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	pm2.61da2ne 3048	Deduction eliminating two inequalities in an antecedent. (Contributed by NM, 29-May-2013.)
⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝜓)    &   ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝜓)    &   ⊢ ((𝜑 ∧ (𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐷)) → 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	pm2.61da3ne 3049	Deduction eliminating three inequalities in an antecedent. (Contributed by NM, 15-Jun-2013.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝜓)    &   ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝜓)    &   ⊢ ((𝜑 ∧ 𝐸 = 𝐹) → 𝜓)    &   ⊢ ((𝜑 ∧ (𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐷 ∧ 𝐸 ≠ 𝐹)) → 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	pm2.61iine 3050	Equality version of pm2.61ii 185. (Contributed by Scott Fenton, 13-Jun-2013.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) → 𝜑)    &   ⊢ (𝐴 = 𝐶 → 𝜑)    &   ⊢ (𝐵 = 𝐷 → 𝜑)    ⇒   ⊢ 𝜑
Theorem	mteqand 3051	A modus tollens deduction for inequality. (Contributed by Steven Nguyen, 1-Jun-2023.)
⊢ (𝜑 → 𝐶 ≠ 𝐷)    &   ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐵)
Theorem	neor 3052	Logical OR with an equality. (Contributed by NM, 29-Apr-2007.)
⊢ ((𝐴 = 𝐵 ∨ 𝜓) ↔ (𝐴 ≠ 𝐵 → 𝜓))
Theorem	neanior 3053	A De Morgan's law for inequality. (Contributed by NM, 18-May-2007.)
⊢ ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐷) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐶 = 𝐷))
Theorem	ne3anior 3054	A De Morgan's law for inequality. (Contributed by NM, 30-Sep-2013.)
⊢ ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐷 ∧ 𝐸 ≠ 𝐹) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐶 = 𝐷 ∨ 𝐸 = 𝐹))
Theorem	neorian 3055	A De Morgan's law for inequality. (Contributed by NM, 18-May-2007.)
⊢ ((𝐴 ≠ 𝐵 ∨ 𝐶 ≠ 𝐷) ↔ ¬ (𝐴 = 𝐵 ∧ 𝐶 = 𝐷))
Theorem	nemtbir 3056	An inference from an inequality, related to modus tollens. (Contributed by NM, 13-Apr-2007.)
⊢ 𝐴 ≠ 𝐵    &   ⊢ (𝜑 ↔ 𝐴 = 𝐵)    ⇒   ⊢ ¬ 𝜑
Theorem	nelne1 3057	Two classes are different if they don't contain the same element. (Contributed by NM, 3-Feb-2012.) (Proof shortened by Wolf Lammen, 14-May-2023.)
⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) → 𝐵 ≠ 𝐶)
Theorem	nelne2 3058	Two classes are different if they don't belong to the same class. (Contributed by NM, 25-Jun-2012.) (Proof shortened by Wolf Lammen, 14-May-2023.)
⊢ ((𝐴 ∈ 𝐶 ∧ ¬ 𝐵 ∈ 𝐶) → 𝐴 ≠ 𝐵)
Theorem	nelelne 3059	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.)
⊢ (¬ 𝐴 ∈ 𝐵 → (𝐶 ∈ 𝐵 → 𝐶 ≠ 𝐴))
Theorem	neneor 3060	If two classes are different, a third class must be different of at least one of them. (Contributed by Thierry Arnoux, 8-Aug-2020.)
⊢ (𝐴 ≠ 𝐵 → (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐶))
Theorem	nfne 3061	Bound-variable hypothesis builder for inequality. (Contributed by NM, 10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥 𝐴 ≠ 𝐵
Theorem	nfned 3062	Bound-variable hypothesis builder for inequality. (Contributed by NM, 10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝐵)    ⇒   ⊢ (𝜑 → Ⅎ𝑥 𝐴 ≠ 𝐵)
Theorem	nabbib 3063	Not equivalent wff's correspond to not equal class abstractions. (Contributed by AV, 7-Apr-2019.) (Proof shortened by Wolf Lammen, 25-Nov-2019.) Definitial form. (Revised by Wolf Lammen, 5-Mar-2025.)
⊢ ({𝑥 ∣ 𝜑} ≠ {𝑥 ∣ 𝜓} ↔ ∃𝑥(𝜑 ↔ ¬ 𝜓))
2.1.4.2  Negated membership
Syntax	wnel 3064	Extend wff notation to include negated membership.
wff 𝐴 ∉ 𝐵
Definition	df-nel 3065	Define negated membership. (Contributed by NM, 7-Aug-1994.)
⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)
Theorem	neli 3066	Inference associated with df-nel 3065. (Contributed by BJ, 7-Jul-2018.)
⊢ 𝐴 ∉ 𝐵    ⇒   ⊢ ¬ 𝐴 ∈ 𝐵
Theorem	nelir 3067	Inference associated with df-nel 3065. (Contributed by BJ, 7-Jul-2018.)
⊢ ¬ 𝐴 ∈ 𝐵    ⇒   ⊢ 𝐴 ∉ 𝐵
Theorem	nelcon3d 3068	Contrapositive law deduction for negated membership. (Contributed by AV, 28-Jan-2020.)
⊢ (𝜑 → (𝐴 ∈ 𝐵 → 𝐶 ∈ 𝐷))    ⇒   ⊢ (𝜑 → (𝐶 ∉ 𝐷 → 𝐴 ∉ 𝐵))
Theorem	neleq12d 3069	Equality theorem for negated membership. (Contributed by FL, 10-Aug-2016.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 ∉ 𝐶 ↔ 𝐵 ∉ 𝐷))
Theorem	neleq1 3070	Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
⊢ (𝐴 = 𝐵 → (𝐴 ∉ 𝐶 ↔ 𝐵 ∉ 𝐶))
Theorem	neleq2 3071	Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
⊢ (𝐴 = 𝐵 → (𝐶 ∉ 𝐴 ↔ 𝐶 ∉ 𝐵))
Theorem	nfnel 3072	Bound-variable hypothesis builder for negated membership. (Contributed by David Abernethy, 26-Jun-2011.) (Revised by Mario Carneiro, 7-Oct-2016.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥 𝐴 ∉ 𝐵
Theorem	nfneld 3073	Bound-variable hypothesis builder for negated membership. (Contributed by David Abernethy, 26-Jun-2011.) (Revised by Mario Carneiro, 7-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝐵)    ⇒   ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∉ 𝐵)
Theorem	nnel 3074	Negation of negated membership, analogous to nne 2964. (Contributed by Alexander van der Vekens, 18-Jan-2018.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
⊢ (¬ 𝐴 ∉ 𝐵 ↔ 𝐴 ∈ 𝐵)
Theorem	elnelne1 3075	Two classes are different if they don't contain the same element. (Contributed by AV, 28-Jan-2020.)
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∉ 𝐶) → 𝐵 ≠ 𝐶)
Theorem	elnelne2 3076	Two classes are different if they don't belong to the same class. (Contributed by AV, 28-Jan-2020.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∉ 𝐶) → 𝐴 ≠ 𝐵)
Theorem	pm2.24nel 3077	A contradiction concerning membership implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
⊢ (𝐴 ∈ 𝐵 → (𝐴 ∉ 𝐵 → 𝜑))
Theorem	pm2.61danel 3078	Deduction eliminating an elementhood in an antecedent. (Contributed by AV, 5-Dec-2021.)
⊢ ((𝜑 ∧ 𝐴 ∈ 𝐵) → 𝜓)    &   ⊢ ((𝜑 ∧ 𝐴 ∉ 𝐵) → 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
2.1.5  Restricted quantification
2.1.5.1  Restricted universal and existential quantification
Syntax	wral 3079	Extend wff notation to include restricted universal quantification.
wff ∀𝑥 ∈ 𝐴 𝜑
Definition	df-ral 3080	Define restricted universal quantification. Special case of Definition 4.15(3) of [TakeutiZaring] p. 22. Note: This notation is most often used to express that 𝜑 holds for all elements of a given class 𝐴. For this reading Ⅎ𝑥𝐴 is required, though, for example, asserted when 𝑥 and 𝐴 are disjoint. Should instead 𝐴 depend on 𝑥, you rather focus on those 𝑥 that happen to be contained in the corresponding 𝐴(𝑥). This hardly used interpretation could still occur naturally. For some examples, look at ralndv1 47697 or ralndv2 47698, courtesy of AV. So be careful to either keep 𝐴 independent of 𝑥, or adjust your comments to include such exotic cases. (Contributed by NM, 19-Aug-1993.)
⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
Theorem	rgen 3081	Generalization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)
⊢ (𝑥 ∈ 𝐴 → 𝜑)    ⇒   ⊢ ∀𝑥 ∈ 𝐴 𝜑
Theorem	ralel 3082	All elements of a class are elements of the class. (Contributed by AV, 30-Oct-2020.)
⊢ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐴
Theorem	rgenw 3083	Generalization rule for restricted quantification. (Contributed by NM, 18-Jun-2014.)
⊢ 𝜑    ⇒   ⊢ ∀𝑥 ∈ 𝐴 𝜑
Theorem	rgen2w 3084	Generalization rule for restricted quantification. Note that 𝑥 and 𝑦 needn't be distinct. (Contributed by NM, 18-Jun-2014.)
⊢ 𝜑    ⇒   ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑
Theorem	mprg 3085	Modus ponens combined with restricted generalization. (Contributed by NM, 10-Aug-2004.)
⊢ (∀𝑥 ∈ 𝐴 𝜑 → 𝜓)    &   ⊢ (𝑥 ∈ 𝐴 → 𝜑)    ⇒   ⊢ 𝜓
Theorem	mprgbir 3086	Modus ponens on biconditional combined with restricted generalization. (Contributed by NM, 21-Mar-2004.)
⊢ (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜓)    &   ⊢ (𝑥 ∈ 𝐴 → 𝜓)    ⇒   ⊢ 𝜑
Theorem	ralrid 3087	Sufficient condition for the restricted universal quantifier. Deduction form. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)
Theorem	raln 3088	Restricted universally quantified negation expressed as a universally quantified negation. (Contributed by BJ, 16-Jul-2021.)
⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ 𝜑))
Syntax	wrex 3089	Extend wff notation to include restricted existential quantification.
wff ∃𝑥 ∈ 𝐴 𝜑
Definition	df-rex 3090	Define restricted existential quantification. Special case of Definition 4.15(4) of [TakeutiZaring] p. 22. Note: This notation is most often used to express that 𝜑 holds for at least one element of a given class 𝐴. For this reading Ⅎ𝑥𝐴 is required, though, for example, asserted when 𝑥 and 𝐴 are disjoint. Should instead 𝐴 depend on 𝑥, you rather assert at least one 𝑥 fulfilling 𝜑 happens to be contained in the corresponding 𝐴(𝑥). This interpretation is rarely needed (see also df-ral 3080). (Contributed by NM, 30-Aug-1993.)
⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
Theorem	ralnex 3091	Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.) (Proof shortened by BJ, 16-Jul-2021.)
⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜑)
Theorem	dfrex2 3092	Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Wolf Lammen, 26-Nov-2019.)
⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑)
Theorem	nrex 3093	Inference adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)
⊢ (𝑥 ∈ 𝐴 → ¬ 𝜓)    ⇒   ⊢ ¬ ∃𝑥 ∈ 𝐴 𝜓
Theorem	alral 3094	Universal quantification implies restricted quantification. (Contributed by NM, 20-Oct-2006.)
⊢ (∀𝑥𝜑 → ∀𝑥 ∈ 𝐴 𝜑)
Theorem	rexex 3095	Restricted existence implies existence. (Contributed by NM, 11-Nov-1995.)
⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥𝜑)
Theorem	rextru 3096	Two ways of expressing that a class has at least one element. (Contributed by Zhi Wang, 23-Sep-2024.)
⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ⊤)
Theorem	ralimi2 3097	Inference quantifying both antecedent and consequent. (Contributed by NM, 22-Feb-2004.)
⊢ ((𝑥 ∈ 𝐴 → 𝜑) → (𝑥 ∈ 𝐵 → 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐵 𝜓)
Theorem	reximi2 3098	Inference quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 8-Nov-2004.)
⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐵 𝜓)
Theorem	ralimia 3099	Inference quantifying both antecedent and consequent. (Contributed by NM, 19-Jul-1996.)
⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜓)
Theorem	reximia 3100	Inference quantifying both antecedent and consequent. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Wolf Lammen, 31-Oct-2024.)
⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)