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

Theoremcleqf 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.)

Theoremabid2f 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.)

Theoremsbabel 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.)

2.1.4  Negated equality and membership

Syntaxwne 2204 Extend wff notation to include inequality.

Syntaxwnel 2205 Extend wff notation to include negated membership.

Definitiondf-ne 2206 Define inequality. (Contributed by NM, 5-Aug-1993.)

Definitiondf-nel 2207 Define negated membership. (Contributed by NM, 7-Aug-1994.)

2.1.4.1  Negated equality

Theoremneii 2208 Inference associated with df-ne 2206. (Contributed by BJ, 7-Jul-2018.)

Theoremneir 2209 Inference associated with df-ne 2206. (Contributed by BJ, 7-Jul-2018.)

Theoremnner 2210 Negation of inequality. (Contributed by Jim Kingdon, 23-Dec-2018.)

Theoremnnedc 2211 Negation of inequality where equality is decidable. (Contributed by Jim Kingdon, 15-May-2018.)
DECID

Theoremdcned 2212 Decidable equality implies decidable negated equality. (Contributed by Jim Kingdon, 3-May-2020.)
DECID        DECID

Theoremneqned 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.)

Theoremneqne 2214 From non equality to inequality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremneirr 2215 No class is unequal to itself. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Proof rewritten by Jim Kingdon, 15-May-2018.)

Theoremdcne 2216 Decidable equality expressed in terms of . Basically the same as df-dc 743. (Contributed by Jim Kingdon, 14-Mar-2020.)
DECID

Theoremnonconne 2217 Law of noncontradiction with equality and inequality. (Contributed by NM, 3-Feb-2012.)

Theoremneeq1 2218 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)

Theoremneeq2 2219 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)

Theoremneeq1i 2220 Inference for inequality. (Contributed by NM, 29-Apr-2005.)

Theoremneeq2i 2221 Inference for inequality. (Contributed by NM, 29-Apr-2005.)

Theoremneeq12i 2222 Inference for inequality. (Contributed by NM, 24-Jul-2012.)

Theoremneeq1d 2223 Deduction for inequality. (Contributed by NM, 25-Oct-1999.)

Theoremneeq2d 2224 Deduction for inequality. (Contributed by NM, 25-Oct-1999.)

Theoremneeq12d 2225 Deduction for inequality. (Contributed by NM, 24-Jul-2012.)

Theoremneneqd 2226 Deduction eliminating inequality definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremneneq 2227 From inequality to non equality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremeqnetri 2228 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Theoremeqnetrd 2229 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Theoremeqnetrri 2230 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Theoremeqnetrrd 2231 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Theoremneeqtri 2232 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Theoremneeqtrd 2233 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Theoremneeqtrri 2234 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Theoremneeqtrrd 2235 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Theoremsyl5eqner 2236 B chained equality inference for inequality. (Contributed by NM, 6-Jun-2012.)

Theorem3netr3d 2237 Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)

Theorem3netr4d 2238 Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)

Theorem3netr3g 2239 Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)

Theorem3netr4g 2240 Substitution of equality into both sides of an inequality. (Contributed by NM, 14-Jun-2012.)

Theoremnecon3abii 2241 Deduction from equality to inequality. (Contributed by NM, 9-Nov-2007.)

Theoremnecon3bbii 2242 Deduction from equality to inequality. (Contributed by NM, 13-Apr-2007.)

Theoremnecon3bii 2243 Inference from equality to inequality. (Contributed by NM, 23-Feb-2005.)

Theoremnecon3abid 2244 Deduction from equality to inequality. (Contributed by NM, 21-Mar-2007.)

Theoremnecon3bbid 2245 Deduction from equality to inequality. (Contributed by NM, 2-Jun-2007.)

Theoremnecon3bid 2246 Deduction from equality to inequality. (Contributed by NM, 23-Feb-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon3ad 2247 Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)

Theoremnecon3bd 2248 Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)

Theoremnecon3d 2249 Contrapositive law deduction for inequality. (Contributed by NM, 10-Jun-2006.)

Theoremnesym 2250 Characterization of inequality in terms of reversed equality (see bicom 128). (Contributed by BJ, 7-Jul-2018.)

Theoremnesymi 2251 Inference associated with nesym 2250. (Contributed by BJ, 7-Jul-2018.)

Theoremnesymir 2252 Inference associated with nesym 2250. (Contributed by BJ, 7-Jul-2018.)

Theoremnecon3i 2253 Contrapositive inference for inequality. (Contributed by NM, 9-Aug-2006.)

Theoremnecon3ai 2254 Contrapositive inference for inequality. (Contributed by NM, 23-May-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)

Theoremnecon3bi 2255 Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)

Theoremnecon1aidc 2256 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 15-May-2018.)
DECID        DECID

Theoremnecon1bidc 2257 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 15-May-2018.)
DECID        DECID

Theoremnecon1idc 2258 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
DECID

Theoremnecon2ai 2259 Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof rewritten by Jim Kingdon, 16-May-2018.)

Theoremnecon2bi 2260 Contrapositive inference for inequality. (Contributed by NM, 1-Apr-2007.)

Theoremnecon2i 2261 Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.)

Theoremnecon2ad 2262 Contrapositive inference for inequality. (Contributed by NM, 19-Apr-2007.) (Proof rewritten by Jim Kingdon, 16-May-2018.)

Theoremnecon2bd 2263 Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)

Theoremnecon2d 2264 Contrapositive inference for inequality. (Contributed by NM, 28-Dec-2008.)

Theoremnecon1abiidc 2265 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
DECID        DECID

Theoremnecon1bbiidc 2266 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
DECID        DECID

Theoremnecon1abiddc 2267 Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
DECID        DECID

Theoremnecon1bbiddc 2268 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
DECID        DECID

Theoremnecon2abiidc 2269 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
DECID        DECID

Theoremnecon2bbii 2270 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
DECID        DECID

Theoremnecon2abiddc 2271 Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
DECID        DECID

Theoremnecon2bbiddc 2272 Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
DECID        DECID

Theoremnecon4aidc 2273 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
DECID        DECID

Theoremnecon4idc 2274 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
DECID        DECID

Theoremnecon4addc 2275 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)
DECID        DECID

Theoremnecon4bddc 2276 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)
DECID        DECID

Theoremnecon4ddc 2277 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)
DECID        DECID

Theoremnecon4abiddc 2278 Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 18-May-2018.)
DECID DECID        DECID DECID

Theoremnecon4bbiddc 2279 Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
DECID DECID        DECID DECID

Theoremnecon4biddc 2280 Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
DECID DECID        DECID DECID

Theoremnecon1addc 2281 Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
DECID        DECID

Theoremnecon1bddc 2282 Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
DECID        DECID

Theoremnecon1ddc 2283 Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
DECID        DECID

Theoremneneqad 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.)

Theoremnebidc 2285 Contraposition law for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
DECID DECID

Theorempm13.18 2286 Theorem *13.18 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)

Theorempm13.181 2287 Theorem *13.181 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)

Theorempm2.21ddne 2288 A contradiction implies anything. Equality/inequality deduction form. (Contributed by David Moews, 28-Feb-2017.)

Theoremnecom 2289 Commutation of inequality. (Contributed by NM, 14-May-1999.)

Theoremnecomi 2290 Inference from commutative law for inequality. (Contributed by NM, 17-Oct-2012.)

Theoremnecomd 2291 Deduction from commutative law for inequality. (Contributed by NM, 12-Feb-2008.)

Theoremneanior 2292 A De Morgan's law for inequality. (Contributed by NM, 18-May-2007.)

Theoremne3anior 2293 A De Morgan's law for inequality. (Contributed by NM, 30-Sep-2013.) (Proof rewritten by Jim Kingdon, 19-May-2018.)

Theoremnemtbir 2294 An inference from an inequality, related to modus tollens. (Contributed by NM, 13-Apr-2007.)

Theoremnelne1 2295 Two classes are different if they don't contain the same element. (Contributed by NM, 3-Feb-2012.)

Theoremnelne2 2296 Two classes are different if they don't belong to the same class. (Contributed by NM, 25-Jun-2012.)

Theoremnfne 2297 Bound-variable hypothesis builder for inequality. (Contributed by NM, 10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremnfned 2298 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 2299 Inference associated with df-nel 2207. (Contributed by BJ, 7-Jul-2018.)

Theoremnelir 2300 Inference associated with df-nel 2207. (Contributed by BJ, 7-Jul-2018.)

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