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

2.1.4  Negated equality and membership

Syntaxwne 2201 Extend wff notation to include inequality.
wff AB

Syntaxwnel 2202 Extend wff notation to include negated membership.
wff AB

Definitiondf-ne 2203 Define inequality. (Contributed by NM, 5-Aug-1993.)
(AB ↔ ¬ A = B)

Definitiondf-nel 2204 Define negated membership. (Contributed by NM, 7-Aug-1994.)
(AB ↔ ¬ A B)

2.1.4.1  Negated equality

Theoremneii 2205 Inference associated with df-ne 2203. (Contributed by BJ, 7-Jul-2018.)
AB        ¬ A = B

Theoremneir 2206 Inference associated with df-ne 2203. (Contributed by BJ, 7-Jul-2018.)
¬ A = B       AB

Theoremnner 2207 Negation of inequality. (Contributed by Jim Kingdon, 23-Dec-2018.)
(A = B → ¬ AB)

Theoremnnedc 2208 Negation of inequality where equality is decidable. (Contributed by Jim Kingdon, 15-May-2018.)
(DECID A = B → (¬ ABA = B))

Theoremdcned 2209 Decidable equality implies decidable negated equality. (Contributed by Jim Kingdon, 3-May-2020.)
(φDECID A = B)       (φDECID AB)

Theoremneirr 2210 No class is unequal to itself. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
¬ AA

Theoremdcne 2211 Decidable equality expressed in terms of . Basically the same as df-dc 742. (Contributed by Jim Kingdon, 14-Mar-2020.)
(DECID A = B ↔ (A = B AB))

Theoremnonconne 2212 Law of noncontradiction with equality and inequality. (Contributed by NM, 3-Feb-2012.)
¬ (A = B AB)

Theoremneeq1 2213 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
(A = B → (A𝐶B𝐶))

Theoremneeq2 2214 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
(A = B → (𝐶A𝐶B))

Theoremneeq1i 2215 Inference for inequality. (Contributed by NM, 29-Apr-2005.)
A = B       (A𝐶B𝐶)

Theoremneeq2i 2216 Inference for inequality. (Contributed by NM, 29-Apr-2005.)
A = B       (𝐶A𝐶B)

Theoremneeq12i 2217 Inference for inequality. (Contributed by NM, 24-Jul-2012.)
A = B    &   𝐶 = 𝐷       (A𝐶B𝐷)

Theoremneeq1d 2218 Deduction for inequality. (Contributed by NM, 25-Oct-1999.)
(φA = B)       (φ → (A𝐶B𝐶))

Theoremneeq2d 2219 Deduction for inequality. (Contributed by NM, 25-Oct-1999.)
(φA = B)       (φ → (𝐶A𝐶B))

Theoremneeq12d 2220 Deduction for inequality. (Contributed by NM, 24-Jul-2012.)
(φA = B)    &   (φ𝐶 = 𝐷)       (φ → (A𝐶B𝐷))

Theoremneneqd 2221 Deduction eliminating inequality definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(φAB)       (φ → ¬ A = B)

Theoremeqnetri 2222 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
A = B    &   B𝐶       A𝐶

Theoremeqnetrd 2223 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
(φA = B)    &   (φB𝐶)       (φA𝐶)

Theoremeqnetrri 2224 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
A = B    &   A𝐶       B𝐶

Theoremeqnetrrd 2225 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
(φA = B)    &   (φA𝐶)       (φB𝐶)

Theoremneeqtri 2226 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
AB    &   B = 𝐶       A𝐶

Theoremneeqtrd 2227 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
(φAB)    &   (φB = 𝐶)       (φA𝐶)

Theoremneeqtrri 2228 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
AB    &   𝐶 = B       A𝐶

Theoremneeqtrrd 2229 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
(φAB)    &   (φ𝐶 = B)       (φA𝐶)

Theoremsyl5eqner 2230 B chained equality inference for inequality. (Contributed by NM, 6-Jun-2012.)
B = A    &   (φB𝐶)       (φA𝐶)

Theorem3netr3d 2231 Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)
(φAB)    &   (φA = 𝐶)    &   (φB = 𝐷)       (φ𝐶𝐷)

Theorem3netr4d 2232 Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)
(φAB)    &   (φ𝐶 = A)    &   (φ𝐷 = B)       (φ𝐶𝐷)

Theorem3netr3g 2233 Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)
(φAB)    &   A = 𝐶    &   B = 𝐷       (φ𝐶𝐷)

Theorem3netr4g 2234 Substitution of equality into both sides of an inequality. (Contributed by NM, 14-Jun-2012.)
(φAB)    &   𝐶 = A    &   𝐷 = B       (φ𝐶𝐷)

Theoremnecon3abii 2235 Deduction from equality to inequality. (Contributed by NM, 9-Nov-2007.)
(A = Bφ)       (AB ↔ ¬ φ)

Theoremnecon3bbii 2236 Deduction from equality to inequality. (Contributed by NM, 13-Apr-2007.)
(φA = B)       φAB)

Theoremnecon3bii 2237 Inference from equality to inequality. (Contributed by NM, 23-Feb-2005.)
(A = B𝐶 = 𝐷)       (AB𝐶𝐷)

Theoremnecon3abid 2238 Deduction from equality to inequality. (Contributed by NM, 21-Mar-2007.)
(φ → (A = Bψ))       (φ → (AB ↔ ¬ ψ))

Theoremnecon3bbid 2239 Deduction from equality to inequality. (Contributed by NM, 2-Jun-2007.)
(φ → (ψA = B))       (φ → (¬ ψAB))

Theoremnecon3bid 2240 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))

Theoremnecon3ad 2241 Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
(φ → (ψA = B))       (φ → (AB → ¬ ψ))

Theoremnecon3bd 2242 Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
(φ → (A = Bψ))       (φ → (¬ ψAB))

Theoremnecon3d 2243 Contrapositive law deduction for inequality. (Contributed by NM, 10-Jun-2006.)
(φ → (A = B𝐶 = 𝐷))       (φ → (𝐶𝐷AB))

Theoremnesym 2244 Characterization of inequality in terms of reversed equality (see bicom 128). (Contributed by BJ, 7-Jul-2018.)
(AB ↔ ¬ B = A)

Theoremnesymi 2245 Inference associated with nesym 2244. (Contributed by BJ, 7-Jul-2018.)
AB        ¬ B = A

Theoremnesymir 2246 Inference associated with nesym 2244. (Contributed by BJ, 7-Jul-2018.)
¬ A = B       BA

Theoremnecon3i 2247 Contrapositive inference for inequality. (Contributed by NM, 9-Aug-2006.)
(A = B𝐶 = 𝐷)       (𝐶𝐷AB)

Theoremnecon3ai 2248 Contrapositive inference for inequality. (Contributed by NM, 23-May-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
(φA = B)       (AB → ¬ φ)

Theoremnecon3bi 2249 Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
(A = Bφ)       φAB)

Theoremnecon1aidc 2250 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 15-May-2018.)
(DECID φ → (¬ φA = B))       (DECID φ → (ABφ))

Theoremnecon1bidc 2251 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 15-May-2018.)
(DECID A = B → (ABφ))       (DECID A = B → (¬ φA = B))

Theoremnecon1idc 2252 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
(AB𝐶 = 𝐷)       (DECID A = B → (𝐶𝐷A = B))

Theoremnecon2ai 2253 Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof rewritten by Jim Kingdon, 16-May-2018.)
(A = B → ¬ φ)       (φAB)

Theoremnecon2bi 2254 Contrapositive inference for inequality. (Contributed by NM, 1-Apr-2007.)
(φAB)       (A = B → ¬ φ)

Theoremnecon2i 2255 Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.)
(A = B𝐶𝐷)       (𝐶 = 𝐷AB)

Theoremnecon2ad 2256 Contrapositive inference for inequality. (Contributed by NM, 19-Apr-2007.) (Proof rewritten by Jim Kingdon, 16-May-2018.)
(φ → (A = B → ¬ ψ))       (φ → (ψAB))

Theoremnecon2bd 2257 Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)
(φ → (ψAB))       (φ → (A = B → ¬ ψ))

Theoremnecon2d 2258 Contrapositive inference for inequality. (Contributed by NM, 28-Dec-2008.)
(φ → (A = B𝐶𝐷))       (φ → (𝐶 = 𝐷AB))

Theoremnecon1abiidc 2259 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
(DECID φ → (¬ φA = B))       (DECID φ → (ABφ))

Theoremnecon1bbiidc 2260 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
(DECID A = B → (ABφ))       (DECID A = B → (¬ φA = B))

Theoremnecon1abiddc 2261 Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
(φ → (DECID ψ → (¬ ψA = B)))       (φ → (DECID ψ → (ABψ)))

Theoremnecon1bbiddc 2262 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
(φ → (DECID A = B → (ABψ)))       (φ → (DECID A = B → (¬ ψA = B)))

Theoremnecon2abiidc 2263 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
(DECID φ → (A = B ↔ ¬ φ))       (DECID φ → (φAB))

Theoremnecon2bbii 2264 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
(DECID A = B → (φAB))       (DECID A = B → (A = B ↔ ¬ φ))

Theoremnecon2abiddc 2265 Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
(φ → (DECID ψ → (A = B ↔ ¬ ψ)))       (φ → (DECID ψ → (ψAB)))

Theoremnecon2bbiddc 2266 Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
(φ → (DECID A = B → (ψAB)))       (φ → (DECID A = B → (A = B ↔ ¬ ψ)))

Theoremnecon4aidc 2267 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
(DECID A = B → (AB → ¬ φ))       (DECID A = B → (φA = B))

Theoremnecon4idc 2268 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
(DECID A = B → (AB𝐶𝐷))       (DECID A = B → (𝐶 = 𝐷A = B))

Theoremnecon4addc 2269 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)
(φ → (DECID A = B → (AB → ¬ ψ)))       (φ → (DECID A = B → (ψA = B)))

Theoremnecon4bddc 2270 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)
(φ → (DECID ψ → (¬ ψAB)))       (φ → (DECID ψ → (A = Bψ)))

Theoremnecon4ddc 2271 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)
(φ → (DECID A = B → (AB𝐶𝐷)))       (φ → (DECID A = B → (𝐶 = 𝐷A = B)))

Theoremnecon4abiddc 2272 Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 18-May-2018.)
(φ → (DECID A = B → (DECID ψ → (AB ↔ ¬ ψ))))       (φ → (DECID A = B → (DECID ψ → (A = Bψ))))

Theoremnecon4bbiddc 2273 Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
(φ → (DECID ψ → (DECID A = B → (¬ ψAB))))       (φ → (DECID ψ → (DECID A = B → (ψA = B))))

Theoremnecon4biddc 2274 Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
(φ → (DECID A = B → (DECID 𝐶 = 𝐷 → (AB𝐶𝐷))))       (φ → (DECID A = B → (DECID 𝐶 = 𝐷 → (A = B𝐶 = 𝐷))))

Theoremnecon1addc 2275 Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
(φ → (DECID ψ → (¬ ψA = B)))       (φ → (DECID ψ → (ABψ)))

Theoremnecon1bddc 2276 Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
(φ → (DECID A = B → (ABψ)))       (φ → (DECID A = B → (¬ ψA = B)))

Theoremnecon1ddc 2277 Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
(φ → (DECID A = B → (AB𝐶 = 𝐷)))       (φ → (DECID A = B → (𝐶𝐷A = B)))

Theoremneneqad 2278 If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2221. One-way deduction form of df-ne 2203. (Contributed by David Moews, 28-Feb-2017.)
(φ → ¬ A = B)       (φAB)

Theoremnebidc 2279 Contraposition law for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
(DECID A = B → (DECID 𝐶 = 𝐷 → ((A = B𝐶 = 𝐷) ↔ (AB𝐶𝐷))))

Theorempm13.18 2280 Theorem *13.18 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
((A = B A𝐶) → B𝐶)

Theorempm13.181 2281 Theorem *13.181 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
((A = B B𝐶) → A𝐶)

Theorempm2.21ddne 2282 A contradiction implies anything. Equality/inequality deduction form. (Contributed by David Moews, 28-Feb-2017.)
(φA = B)    &   (φAB)       (φψ)

Theoremnecom 2283 Commutation of inequality. (Contributed by NM, 14-May-1999.)
(ABBA)

Theoremnecomi 2284 Inference from commutative law for inequality. (Contributed by NM, 17-Oct-2012.)
AB       BA

Theoremnecomd 2285 Deduction from commutative law for inequality. (Contributed by NM, 12-Feb-2008.)
(φAB)       (φBA)

Theoremneanior 2286 A De Morgan's law for inequality. (Contributed by NM, 18-May-2007.)
((AB 𝐶𝐷) ↔ ¬ (A = B 𝐶 = 𝐷))

Theoremne3anior 2287 A De Morgan's law for inequality. (Contributed by NM, 30-Sep-2013.) (Proof rewritten by Jim Kingdon, 19-May-2018.)
((AB 𝐶𝐷 𝐸𝐹) ↔ ¬ (A = B 𝐶 = 𝐷 𝐸 = 𝐹))

Theoremnemtbir 2288 An inference from an inequality, related to modus tollens. (Contributed by NM, 13-Apr-2007.)
AB    &   (φA = B)        ¬ φ

Theoremnelne1 2289 Two classes are different if they don't contain the same element. (Contributed by NM, 3-Feb-2012.)
((A B ¬ A 𝐶) → B𝐶)

Theoremnelne2 2290 Two classes are different if they don't belong to the same class. (Contributed by NM, 25-Jun-2012.)
((A 𝐶 ¬ B 𝐶) → AB)

Theoremnfne 2291 Bound-variable hypothesis builder for inequality. (Contributed by NM, 10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.)
xA    &   xB       x AB

Theoremnfned 2292 Bound-variable hypothesis builder for inequality. (Contributed by NM, 10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.)
(φxA)    &   (φxB)       (φ → Ⅎx AB)

2.1.4.2  Negated membership

Theoremneli 2293 Inference associated with df-nel 2204. (Contributed by BJ, 7-Jul-2018.)
AB        ¬ A B

Theoremnelir 2294 Inference associated with df-nel 2204. (Contributed by BJ, 7-Jul-2018.)
¬ A B       AB

Theoremneleq1 2295 Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.)
(A = B → (A𝐶B𝐶))

Theoremneleq2 2296 Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.)
(A = B → (𝐶A𝐶B))

Theoremneleq12d 2297 Equality theorem for negated membership. (Contributed by FL, 10-Aug-2016.)
(φA = B)    &   (φ𝐶 = 𝐷)       (φ → (A𝐶B𝐷))

Theoremnfnel 2298 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 2299 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)

2.1.5  Restricted quantification

Syntaxwral 2300 Extend wff notation to include restricted universal quantification.
wff x A φ

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