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Theorem List for Intuitionistic Logic Explorer - 2301-2400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdrnfc2 2301 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 8-Oct-2016.)

Theoremnfabd 2302 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 8-Oct-2016.)

Theoremdvelimdc 2303 Deduction form of dvelimc 2304. (Contributed by Mario Carneiro, 8-Oct-2016.)

Theoremdvelimc 2304 Version of dvelim 1985 for classes. (Contributed by Mario Carneiro, 8-Oct-2016.)

Theoremnfcvf 2305 If and are distinct, then is not free in . (Contributed by Mario Carneiro, 8-Oct-2016.)

Theoremnfcvf2 2306 If and are distinct, then is not free in . (Contributed by Mario Carneiro, 5-Dec-2016.)

Theoremcleqf 2307 Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2241. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremabid2f 2308 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 2309* 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

2.1.4.1  Negated equality

Syntaxwne 2310 Extend wff notation to include inequality.

Definitiondf-ne 2311 Define inequality. (Contributed by NM, 5-Aug-1993.)

Theoremneii 2312 Inference associated with df-ne 2311. (Contributed by BJ, 7-Jul-2018.)

Theoremneir 2313 Inference associated with df-ne 2311. (Contributed by BJ, 7-Jul-2018.)

Theoremnner 2314 Negation of inequality. (Contributed by Jim Kingdon, 23-Dec-2018.)

Theoremnnedc 2315 Negation of inequality where equality is decidable. (Contributed by Jim Kingdon, 15-May-2018.)
DECID

Theoremdcned 2316 Decidable equality implies decidable negated equality. (Contributed by Jim Kingdon, 3-May-2020.)
DECID        DECID

Theoremneqned 2317 If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2331. One-way deduction form of df-ne 2311. (Contributed by David Moews, 28-Feb-2017.) Allow a shortening of necon3bi 2360. (Revised by Wolf Lammen, 22-Nov-2019.)

Theoremneqne 2318 From non-equality to inequality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremneirr 2319 No class is unequal to itself. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Proof rewritten by Jim Kingdon, 15-May-2018.)

Theoremeqneqall 2320 A contradiction concerning equality implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.)

Theoremdcne 2321 Decidable equality expressed in terms of . Basically the same as df-dc 821. (Contributed by Jim Kingdon, 14-Mar-2020.)
DECID

Theoremnonconne 2322 Law of noncontradiction with equality and inequality. (Contributed by NM, 3-Feb-2012.)

Theoremneeq1 2323 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)

Theoremneeq2 2324 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)

Theoremneeq1i 2325 Inference for inequality. (Contributed by NM, 29-Apr-2005.)

Theoremneeq2i 2326 Inference for inequality. (Contributed by NM, 29-Apr-2005.)

Theoremneeq12i 2327 Inference for inequality. (Contributed by NM, 24-Jul-2012.)

Theoremneeq1d 2328 Deduction for inequality. (Contributed by NM, 25-Oct-1999.)

Theoremneeq2d 2329 Deduction for inequality. (Contributed by NM, 25-Oct-1999.)

Theoremneeq12d 2330 Deduction for inequality. (Contributed by NM, 24-Jul-2012.)

Theoremneneqd 2331 Deduction eliminating inequality definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremneneq 2332 From inequality to non-equality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremeqnetri 2333 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Theoremeqnetrd 2334 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Theoremeqnetrri 2335 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Theoremeqnetrrd 2336 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Theoremneeqtri 2337 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Theoremneeqtrd 2338 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Theoremneeqtrri 2339 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Theoremneeqtrrd 2340 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Theoremeqnetrrid 2341 B chained equality inference for inequality. (Contributed by NM, 6-Jun-2012.)

Theorem3netr3d 2342 Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)

Theorem3netr4d 2343 Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)

Theorem3netr3g 2344 Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)

Theorem3netr4g 2345 Substitution of equality into both sides of an inequality. (Contributed by NM, 14-Jun-2012.)

Theoremnecon3abii 2346 Deduction from equality to inequality. (Contributed by NM, 9-Nov-2007.)

Theoremnecon3bbii 2347 Deduction from equality to inequality. (Contributed by NM, 13-Apr-2007.)

Theoremnecon3bii 2348 Inference from equality to inequality. (Contributed by NM, 23-Feb-2005.)

Theoremnecon3abid 2349 Deduction from equality to inequality. (Contributed by NM, 21-Mar-2007.)

Theoremnecon3bbid 2350 Deduction from equality to inequality. (Contributed by NM, 2-Jun-2007.)

Theoremnecon3bid 2351 Deduction from equality to inequality. (Contributed by NM, 23-Feb-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon3ad 2352 Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)

Theoremnecon3bd 2353 Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)

Theoremnecon3d 2354 Contrapositive law deduction for inequality. (Contributed by NM, 10-Jun-2006.)

Theoremnesym 2355 Characterization of inequality in terms of reversed equality (see bicom 139). (Contributed by BJ, 7-Jul-2018.)

Theoremnesymi 2356 Inference associated with nesym 2355. (Contributed by BJ, 7-Jul-2018.)

Theoremnesymir 2357 Inference associated with nesym 2355. (Contributed by BJ, 7-Jul-2018.)

Theoremnecon3i 2358 Contrapositive inference for inequality. (Contributed by NM, 9-Aug-2006.)

Theoremnecon3ai 2359 Contrapositive inference for inequality. (Contributed by NM, 23-May-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)

Theoremnecon3bi 2360 Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)

Theoremnecon1aidc 2361 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 15-May-2018.)
DECID        DECID

Theoremnecon1bidc 2362 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 15-May-2018.)
DECID        DECID

Theoremnecon1idc 2363 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
DECID

Theoremnecon2ai 2364 Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof rewritten by Jim Kingdon, 16-May-2018.)

Theoremnecon2bi 2365 Contrapositive inference for inequality. (Contributed by NM, 1-Apr-2007.)

Theoremnecon2i 2366 Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.)

Theoremnecon2ad 2367 Contrapositive inference for inequality. (Contributed by NM, 19-Apr-2007.) (Proof rewritten by Jim Kingdon, 16-May-2018.)

Theoremnecon2bd 2368 Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)

Theoremnecon2d 2369 Contrapositive inference for inequality. (Contributed by NM, 28-Dec-2008.)

Theoremnecon1abiidc 2370 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
DECID        DECID

Theoremnecon1bbiidc 2371 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
DECID        DECID

Theoremnecon1abiddc 2372 Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
DECID        DECID

Theoremnecon1bbiddc 2373 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
DECID        DECID

Theoremnecon2abiidc 2374 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
DECID        DECID

Theoremnecon2bbiidc 2375 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
DECID        DECID

Theoremnecon2abiddc 2376 Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
DECID        DECID

Theoremnecon2bbiddc 2377 Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
DECID        DECID

Theoremnecon4aidc 2378 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
DECID        DECID

Theoremnecon4idc 2379 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
DECID        DECID

Theoremnecon4addc 2380 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)
DECID        DECID

Theoremnecon4bddc 2381 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)
DECID        DECID

Theoremnecon4ddc 2382 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)
DECID        DECID

Theoremnecon4abiddc 2383 Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 18-May-2018.)
DECID DECID        DECID DECID

Theoremnecon4bbiddc 2384 Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
DECID DECID        DECID DECID

Theoremnecon4biddc 2385 Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
DECID DECID        DECID DECID

Theoremnecon1addc 2386 Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
DECID        DECID

Theoremnecon1bddc 2387 Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
DECID        DECID

Theoremnecon1ddc 2388 Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
DECID        DECID

Theoremneneqad 2389 If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2331. One-way deduction form of df-ne 2311. (Contributed by David Moews, 28-Feb-2017.)

Theoremnebidc 2390 Contraposition law for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
DECID DECID

Theorempm13.18 2391 Theorem *13.18 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)

Theorempm13.181 2392 Theorem *13.181 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)

Theorempm2.21ddne 2393 A contradiction implies anything. Equality/inequality deduction form. (Contributed by David Moews, 28-Feb-2017.)

Theoremnecom 2394 Commutation of inequality. (Contributed by NM, 14-May-1999.)

Theoremnecomi 2395 Inference from commutative law for inequality. (Contributed by NM, 17-Oct-2012.)

Theoremnecomd 2396 Deduction from commutative law for inequality. (Contributed by NM, 12-Feb-2008.)

Theoremneanior 2397 A De Morgan's law for inequality. (Contributed by NM, 18-May-2007.)

Theoremne3anior 2398 A De Morgan's law for inequality. (Contributed by NM, 30-Sep-2013.) (Proof rewritten by Jim Kingdon, 19-May-2018.)

Theoremnemtbir 2399 An inference from an inequality, related to modus tollens. (Contributed by NM, 13-Apr-2007.)

Theoremnelne1 2400 Two classes are different if they don't contain the same element. (Contributed by NM, 3-Feb-2012.)

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