P. 23 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 2201-2300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cleqf 2201	Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2137. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
Theorem	abid2f 2202	A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35. (Contributed by NM, 5-Sep-2011.) (Revised by Mario Carneiro, 7-Oct-2016.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴
Theorem	sbabel 2203*	Theorem to move a substitution in and out of a class abstraction. (Contributed by NM, 27-Sep-2003.) (Revised by Mario Carneiro, 7-Oct-2016.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ ([𝑦 / 𝑥]{𝑧 ∣ 𝜑} ∈ 𝐴 ↔ {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∈ 𝐴)
2.1.4  Negated equality and membership
Syntax	wne 2204	Extend wff notation to include inequality.
wff 𝐴 ≠ 𝐵
Syntax	wnel 2205	Extend wff notation to include negated membership.
wff 𝐴 ∉ 𝐵
Definition	df-ne 2206	Define inequality. (Contributed by NM, 5-Aug-1993.)
⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
Definition	df-nel 2207	Define negated membership. (Contributed by NM, 7-Aug-1994.)
⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)
2.1.4.1  Negated equality
Theorem	neii 2208	Inference associated with df-ne 2206. (Contributed by BJ, 7-Jul-2018.)
⊢ 𝐴 ≠ 𝐵    ⇒   ⊢ ¬ 𝐴 = 𝐵
Theorem	neir 2209	Inference associated with df-ne 2206. (Contributed by BJ, 7-Jul-2018.)
⊢ ¬ 𝐴 = 𝐵    ⇒   ⊢ 𝐴 ≠ 𝐵
Theorem	nner 2210	Negation of inequality. (Contributed by Jim Kingdon, 23-Dec-2018.)
⊢ (𝐴 = 𝐵 → ¬ 𝐴 ≠ 𝐵)
Theorem	nnedc 2211	Negation of inequality where equality is decidable. (Contributed by Jim Kingdon, 15-May-2018.)
⊢ (DECID 𝐴 = 𝐵 → (¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵))
Theorem	dcned 2212	Decidable equality implies decidable negated equality. (Contributed by Jim Kingdon, 3-May-2020.)
⊢ (𝜑 → DECID 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → DECID 𝐴 ≠ 𝐵)
Theorem	neqned 2213	If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2226. One-way deduction form of df-ne 2206. (Contributed by David Moews, 28-Feb-2017.) Allow a shortening of necon3bi 2255. (Revised by Wolf Lammen, 22-Nov-2019.)
⊢ (𝜑 → ¬ 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐵)
Theorem	neqne 2214	From non equality to inequality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (¬ 𝐴 = 𝐵 → 𝐴 ≠ 𝐵)
Theorem	neirr 2215	No class is unequal to itself. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
⊢ ¬ 𝐴 ≠ 𝐴
Theorem	dcne 2216	Decidable equality expressed in terms of ≠. Basically the same as df-dc 743. (Contributed by Jim Kingdon, 14-Mar-2020.)
⊢ (DECID 𝐴 = 𝐵 ↔ (𝐴 = 𝐵 ∨ 𝐴 ≠ 𝐵))
Theorem	nonconne 2217	Law of noncontradiction with equality and inequality. (Contributed by NM, 3-Feb-2012.)
⊢ ¬ (𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐵)
Theorem	neeq1 2218	Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶))
Theorem	neeq2 2219	Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
⊢ (𝐴 = 𝐵 → (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵))
Theorem	neeq1i 2220	Inference for inequality. (Contributed by NM, 29-Apr-2005.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶)
Theorem	neeq2i 2221	Inference for inequality. (Contributed by NM, 29-Apr-2005.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵)
Theorem	neeq12i 2222	Inference for inequality. (Contributed by NM, 24-Jul-2012.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐷)
Theorem	neeq1d 2223	Deduction for inequality. (Contributed by NM, 25-Oct-1999.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶))
Theorem	neeq2d 2224	Deduction for inequality. (Contributed by NM, 25-Oct-1999.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵))
Theorem	neeq12d 2225	Deduction for inequality. (Contributed by NM, 24-Jul-2012.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐷))
Theorem	neneqd 2226	Deduction eliminating inequality definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ (𝜑 → 𝐴 ≠ 𝐵)    ⇒   ⊢ (𝜑 → ¬ 𝐴 = 𝐵)
Theorem	neneq 2227	From inequality to non equality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝐴 ≠ 𝐵 → ¬ 𝐴 = 𝐵)
Theorem	eqnetri 2228	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐵 ≠ 𝐶    ⇒   ⊢ 𝐴 ≠ 𝐶
Theorem	eqnetrd 2229	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐵 ≠ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐶)
Theorem	eqnetrri 2230	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐴 ≠ 𝐶    ⇒   ⊢ 𝐵 ≠ 𝐶
Theorem	eqnetrrd 2231	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐶)    ⇒   ⊢ (𝜑 → 𝐵 ≠ 𝐶)
Theorem	neeqtri 2232	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
⊢ 𝐴 ≠ 𝐵    &   ⊢ 𝐵 = 𝐶    ⇒   ⊢ 𝐴 ≠ 𝐶
Theorem	neeqtrd 2233	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐶)
Theorem	neeqtrri 2234	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
⊢ 𝐴 ≠ 𝐵    &   ⊢ 𝐶 = 𝐵    ⇒   ⊢ 𝐴 ≠ 𝐶
Theorem	neeqtrrd 2235	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐶)
Theorem	syl5eqner 2236	B chained equality inference for inequality. (Contributed by NM, 6-Jun-2012.)
⊢ 𝐵 = 𝐴    &   ⊢ (𝜑 → 𝐵 ≠ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐶)
Theorem	3netr3d 2237	Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)
⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → 𝐴 = 𝐶)    &   ⊢ (𝜑 → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → 𝐶 ≠ 𝐷)
Theorem	3netr4d 2238	Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)
⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐴)    &   ⊢ (𝜑 → 𝐷 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐶 ≠ 𝐷)
Theorem	3netr3g 2239	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 2240	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	necon3abii 2241	Deduction from equality to inequality. (Contributed by NM, 9-Nov-2007.)
⊢ (𝐴 = 𝐵 ↔ 𝜑)    ⇒   ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝜑)
Theorem	necon3bbii 2242	Deduction from equality to inequality. (Contributed by NM, 13-Apr-2007.)
⊢ (𝜑 ↔ 𝐴 = 𝐵)    ⇒   ⊢ (¬ 𝜑 ↔ 𝐴 ≠ 𝐵)
Theorem	necon3bii 2243	Inference from equality to inequality. (Contributed by NM, 23-Feb-2005.)
⊢ (𝐴 = 𝐵 ↔ 𝐶 = 𝐷)    ⇒   ⊢ (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷)
Theorem	necon3abid 2244	Deduction from equality to inequality. (Contributed by NM, 21-Mar-2007.)
⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝜓))    ⇒   ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ ¬ 𝜓))
Theorem	necon3bbid 2245	Deduction from equality to inequality. (Contributed by NM, 2-Jun-2007.)
⊢ (𝜑 → (𝜓 ↔ 𝐴 = 𝐵))    ⇒   ⊢ (𝜑 → (¬ 𝜓 ↔ 𝐴 ≠ 𝐵))
Theorem	necon3bid 2246	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	necon3ad 2247	Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
⊢ (𝜑 → (𝜓 → 𝐴 = 𝐵))    ⇒   ⊢ (𝜑 → (𝐴 ≠ 𝐵 → ¬ 𝜓))
Theorem	necon3bd 2248	Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
⊢ (𝜑 → (𝐴 = 𝐵 → 𝜓))    ⇒   ⊢ (𝜑 → (¬ 𝜓 → 𝐴 ≠ 𝐵))
Theorem	necon3d 2249	Contrapositive law deduction for inequality. (Contributed by NM, 10-Jun-2006.)
⊢ (𝜑 → (𝐴 = 𝐵 → 𝐶 = 𝐷))    ⇒   ⊢ (𝜑 → (𝐶 ≠ 𝐷 → 𝐴 ≠ 𝐵))
Theorem	nesym 2250	Characterization of inequality in terms of reversed equality (see bicom 128). (Contributed by BJ, 7-Jul-2018.)
⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐵 = 𝐴)
Theorem	nesymi 2251	Inference associated with nesym 2250. (Contributed by BJ, 7-Jul-2018.)
⊢ 𝐴 ≠ 𝐵    ⇒   ⊢ ¬ 𝐵 = 𝐴
Theorem	nesymir 2252	Inference associated with nesym 2250. (Contributed by BJ, 7-Jul-2018.)
⊢ ¬ 𝐴 = 𝐵    ⇒   ⊢ 𝐵 ≠ 𝐴
Theorem	necon3i 2253	Contrapositive inference for inequality. (Contributed by NM, 9-Aug-2006.)
⊢ (𝐴 = 𝐵 → 𝐶 = 𝐷)    ⇒   ⊢ (𝐶 ≠ 𝐷 → 𝐴 ≠ 𝐵)
Theorem	necon3ai 2254	Contrapositive inference for inequality. (Contributed by NM, 23-May-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝐴 ≠ 𝐵 → ¬ 𝜑)
Theorem	necon3bi 2255	Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
⊢ (𝐴 = 𝐵 → 𝜑)    ⇒   ⊢ (¬ 𝜑 → 𝐴 ≠ 𝐵)
Theorem	necon1aidc 2256	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 15-May-2018.)
⊢ (DECID 𝜑 → (¬ 𝜑 → 𝐴 = 𝐵))    ⇒   ⊢ (DECID 𝜑 → (𝐴 ≠ 𝐵 → 𝜑))
Theorem	necon1bidc 2257	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 15-May-2018.)
⊢ (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → 𝜑))    ⇒   ⊢ (DECID 𝐴 = 𝐵 → (¬ 𝜑 → 𝐴 = 𝐵))
Theorem	necon1idc 2258	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
⊢ (𝐴 ≠ 𝐵 → 𝐶 = 𝐷)    ⇒   ⊢ (DECID 𝐴 = 𝐵 → (𝐶 ≠ 𝐷 → 𝐴 = 𝐵))
Theorem	necon2ai 2259	Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof rewritten by Jim Kingdon, 16-May-2018.)
⊢ (𝐴 = 𝐵 → ¬ 𝜑)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐵)
Theorem	necon2bi 2260	Contrapositive inference for inequality. (Contributed by NM, 1-Apr-2007.)
⊢ (𝜑 → 𝐴 ≠ 𝐵)    ⇒   ⊢ (𝐴 = 𝐵 → ¬ 𝜑)
Theorem	necon2i 2261	Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.)
⊢ (𝐴 = 𝐵 → 𝐶 ≠ 𝐷)    ⇒   ⊢ (𝐶 = 𝐷 → 𝐴 ≠ 𝐵)
Theorem	necon2ad 2262	Contrapositive inference for inequality. (Contributed by NM, 19-Apr-2007.) (Proof rewritten by Jim Kingdon, 16-May-2018.)
⊢ (𝜑 → (𝐴 = 𝐵 → ¬ 𝜓))    ⇒   ⊢ (𝜑 → (𝜓 → 𝐴 ≠ 𝐵))
Theorem	necon2bd 2263	Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)
⊢ (𝜑 → (𝜓 → 𝐴 ≠ 𝐵))    ⇒   ⊢ (𝜑 → (𝐴 = 𝐵 → ¬ 𝜓))
Theorem	necon2d 2264	Contrapositive inference for inequality. (Contributed by NM, 28-Dec-2008.)
⊢ (𝜑 → (𝐴 = 𝐵 → 𝐶 ≠ 𝐷))    ⇒   ⊢ (𝜑 → (𝐶 = 𝐷 → 𝐴 ≠ 𝐵))
Theorem	necon1abiidc 2265	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
⊢ (DECID 𝜑 → (¬ 𝜑 ↔ 𝐴 = 𝐵))    ⇒   ⊢ (DECID 𝜑 → (𝐴 ≠ 𝐵 ↔ 𝜑))
Theorem	necon1bbiidc 2266	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
⊢ (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 ↔ 𝜑))    ⇒   ⊢ (DECID 𝐴 = 𝐵 → (¬ 𝜑 ↔ 𝐴 = 𝐵))
Theorem	necon1abiddc 2267	Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜓 ↔ 𝐴 = 𝐵)))    ⇒   ⊢ (𝜑 → (DECID 𝜓 → (𝐴 ≠ 𝐵 ↔ 𝜓)))
Theorem	necon1bbiddc 2268	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 ↔ 𝜓)))    ⇒   ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (¬ 𝜓 ↔ 𝐴 = 𝐵)))
Theorem	necon2abiidc 2269	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
⊢ (DECID 𝜑 → (𝐴 = 𝐵 ↔ ¬ 𝜑))    ⇒   ⊢ (DECID 𝜑 → (𝜑 ↔ 𝐴 ≠ 𝐵))
Theorem	necon2bbii 2270	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
⊢ (DECID 𝐴 = 𝐵 → (𝜑 ↔ 𝐴 ≠ 𝐵))    ⇒   ⊢ (DECID 𝐴 = 𝐵 → (𝐴 = 𝐵 ↔ ¬ 𝜑))
Theorem	necon2abiddc 2271	Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
⊢ (𝜑 → (DECID 𝜓 → (𝐴 = 𝐵 ↔ ¬ 𝜓)))    ⇒   ⊢ (𝜑 → (DECID 𝜓 → (𝜓 ↔ 𝐴 ≠ 𝐵)))
Theorem	necon2bbiddc 2272	Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝜓 ↔ 𝐴 ≠ 𝐵)))    ⇒   ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 = 𝐵 ↔ ¬ 𝜓)))
Theorem	necon4aidc 2273	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
⊢ (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → ¬ 𝜑))    ⇒   ⊢ (DECID 𝐴 = 𝐵 → (𝜑 → 𝐴 = 𝐵))
Theorem	necon4idc 2274	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
⊢ (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → 𝐶 ≠ 𝐷))    ⇒   ⊢ (DECID 𝐴 = 𝐵 → (𝐶 = 𝐷 → 𝐴 = 𝐵))
Theorem	necon4addc 2275	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)
⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → ¬ 𝜓)))    ⇒   ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝜓 → 𝐴 = 𝐵)))
Theorem	necon4bddc 2276	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)
⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜓 → 𝐴 ≠ 𝐵)))    ⇒   ⊢ (𝜑 → (DECID 𝜓 → (𝐴 = 𝐵 → 𝜓)))
Theorem	necon4ddc 2277	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)
⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → 𝐶 ≠ 𝐷)))    ⇒   ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐶 = 𝐷 → 𝐴 = 𝐵)))
Theorem	necon4abiddc 2278	Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 18-May-2018.)
⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝜓 → (𝐴 ≠ 𝐵 ↔ ¬ 𝜓))))    ⇒   ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝜓 → (𝐴 = 𝐵 ↔ 𝜓))))
Theorem	necon4bbiddc 2279	Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
⊢ (𝜑 → (DECID 𝜓 → (DECID 𝐴 = 𝐵 → (¬ 𝜓 ↔ 𝐴 ≠ 𝐵))))    ⇒   ⊢ (𝜑 → (DECID 𝜓 → (DECID 𝐴 = 𝐵 → (𝜓 ↔ 𝐴 = 𝐵))))
Theorem	necon4biddc 2280	Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝐶 = 𝐷 → (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷))))    ⇒   ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝐶 = 𝐷 → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷))))
Theorem	necon1addc 2281	Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜓 → 𝐴 = 𝐵)))    ⇒   ⊢ (𝜑 → (DECID 𝜓 → (𝐴 ≠ 𝐵 → 𝜓)))
Theorem	necon1bddc 2282	Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → 𝜓)))    ⇒   ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (¬ 𝜓 → 𝐴 = 𝐵)))
Theorem	necon1ddc 2283	Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → 𝐶 = 𝐷)))    ⇒   ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐶 ≠ 𝐷 → 𝐴 = 𝐵)))
Theorem	neneqad 2284	If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2226. One-way deduction form of df-ne 2206. (Contributed by David Moews, 28-Feb-2017.)
⊢ (𝜑 → ¬ 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐵)
Theorem	nebidc 2285	Contraposition law for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
⊢ (DECID 𝐴 = 𝐵 → (DECID 𝐶 = 𝐷 → ((𝐴 = 𝐵 ↔ 𝐶 = 𝐷) ↔ (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷))))
Theorem	pm13.18 2286	Theorem *13.18 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
⊢ ((𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐶) → 𝐵 ≠ 𝐶)
Theorem	pm13.181 2287	Theorem *13.181 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
⊢ ((𝐴 = 𝐵 ∧ 𝐵 ≠ 𝐶) → 𝐴 ≠ 𝐶)
Theorem	pm2.21ddne 2288	A contradiction implies anything. Equality/inequality deduction form. (Contributed by David Moews, 28-Feb-2017.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	necom 2289	Commutation of inequality. (Contributed by NM, 14-May-1999.)
⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴)
Theorem	necomi 2290	Inference from commutative law for inequality. (Contributed by NM, 17-Oct-2012.)
⊢ 𝐴 ≠ 𝐵    ⇒   ⊢ 𝐵 ≠ 𝐴
Theorem	necomd 2291	Deduction from commutative law for inequality. (Contributed by NM, 12-Feb-2008.)
⊢ (𝜑 → 𝐴 ≠ 𝐵)    ⇒   ⊢ (𝜑 → 𝐵 ≠ 𝐴)
Theorem	neanior 2292	A De Morgan's law for inequality. (Contributed by NM, 18-May-2007.)
⊢ ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐷) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐶 = 𝐷))
Theorem	ne3anior 2293	A De Morgan's law for inequality. (Contributed by NM, 30-Sep-2013.) (Proof rewritten by Jim Kingdon, 19-May-2018.)
⊢ ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐷 ∧ 𝐸 ≠ 𝐹) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐶 = 𝐷 ∨ 𝐸 = 𝐹))
Theorem	nemtbir 2294	An inference from an inequality, related to modus tollens. (Contributed by NM, 13-Apr-2007.)
⊢ 𝐴 ≠ 𝐵    &   ⊢ (𝜑 ↔ 𝐴 = 𝐵)    ⇒   ⊢ ¬ 𝜑
Theorem	nelne1 2295	Two classes are different if they don't contain the same element. (Contributed by NM, 3-Feb-2012.)
⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) → 𝐵 ≠ 𝐶)
Theorem	nelne2 2296	Two classes are different if they don't belong to the same class. (Contributed by NM, 25-Jun-2012.)
⊢ ((𝐴 ∈ 𝐶 ∧ ¬ 𝐵 ∈ 𝐶) → 𝐴 ≠ 𝐵)
Theorem	nfne 2297	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 2298	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)
2.1.4.2  Negated membership
Theorem	neli 2299	Inference associated with df-nel 2207. (Contributed by BJ, 7-Jul-2018.)
⊢ 𝐴 ∉ 𝐵    ⇒   ⊢ ¬ 𝐴 ∈ 𝐵
Theorem	nelir 2300	Inference associated with df-nel 2207. (Contributed by BJ, 7-Jul-2018.)
⊢ ¬ 𝐴 ∈ 𝐵    ⇒   ⊢ 𝐴 ∉ 𝐵