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Theorem List for Metamath Proof Explorer - 2401-2500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdrnfc1 2401 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 8-Oct-2016.)

Theoremdrnfc2 2402 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 8-Oct-2016.)

Theoremnfabd2 2403 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 8-Oct-2016.)

Theoremnfabd 2404 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 8-Oct-2016.)

Theoremdvelimdc 2405 Deduction form of dvelimc 2406. (Contributed by Mario Carneiro, 8-Oct-2016.)

Theoremdvelimc 2406 Version of dvelim 2092 for classes. (Contributed by Mario Carneiro, 8-Oct-2016.)

Theoremnfcvf 2407 If and are distinct, then is not free in . (Contributed by Mario Carneiro, 8-Oct-2016.)

Theoremnfcvf2 2408 If and are distinct, then is not free in . (Contributed by Mario Carneiro, 5-Dec-2016.)

Theoremcleqf 2409 Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremabid2f 2410 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 2411* 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 2412 Extend wff notation to include inequality.

Syntaxwnel 2413 Extend wff notation to include negated membership.

Definitiondf-ne 2414 Define inequality. (Contributed by NM, 5-Aug-1993.)

Definitiondf-nel 2415 Define negated membership. (Contributed by NM, 7-Aug-1994.)

Theoremnne 2416 Negation of inequality. (Contributed by NM, 9-Jun-2006.)

Theoremneirr 2417 No class is unequal to itself. (Contributed by Stefan O'Rear, 1-Jan-2015.)

Theoremexmidne 2418 Excluded middle with equality and inequality. (Contributed by NM, 3-Feb-2012.)

Theoremnonconne 2419 Law of noncontradiction with equality and inequality. (Contributed by NM, 3-Feb-2012.)

Theoremneeq1 2420 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)

Theoremneeq2 2421 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)

Theoremneeq1i 2422 Inference for inequality. (Contributed by NM, 29-Apr-2005.)

Theoremneeq2i 2423 Inference for inequality. (Contributed by NM, 29-Apr-2005.)

Theoremneeq12i 2424 Inference for inequality. (Contributed by NM, 24-Jul-2012.)

Theoremneeq1d 2425 Deduction for inequality. (Contributed by NM, 25-Oct-1999.)

Theoremneeq2d 2426 Deduction for inequality. (Contributed by NM, 25-Oct-1999.)

Theoremneeq12d 2427 Deduction for inequality. (Contributed by NM, 24-Jul-2012.)

Theoremneneqd 2428 Deduction eliminating inequality definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremeqnetri 2429 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Theoremeqnetrd 2430 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Theoremeqnetrri 2431 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Theoremeqnetrrd 2432 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Theoremneeqtri 2433 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Theoremneeqtrd 2434 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Theoremneeqtrri 2435 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Theoremneeqtrrd 2436 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Theoremsyl5eqner 2437 B chained equality inference for inequality. (Contributed by NM, 6-Jun-2012.)

Theorem3netr3d 2438 Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)

Theorem3netr4d 2439 Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)

Theorem3netr3g 2440 Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)

Theorem3netr4g 2441 Substitution of equality into both sides of an inequality. (Contributed by NM, 14-Jun-2012.)

Theoremnecon3abii 2442 Deduction from equality to inequality. (Contributed by NM, 9-Nov-2007.)

Theoremnecon3bbii 2443 Deduction from equality to inequality. (Contributed by NM, 13-Apr-2007.)

Theoremnecon3bii 2444 Inference from equality to inequality. (Contributed by NM, 23-Feb-2005.)

Theoremnecon3abid 2445 Deduction from equality to inequality. (Contributed by NM, 21-Mar-2007.)

Theoremnecon3bbid 2446 Deduction from equality to inequality. (Contributed by NM, 2-Jun-2007.)

Theoremnecon3bid 2447 Deduction from equality to inequality. (Contributed by NM, 23-Feb-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon3ad 2448 Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon3bd 2449 Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon3d 2450 Contrapositive law deduction for inequality. (Contributed by NM, 10-Jun-2006.)

Theoremnecon3i 2451 Contrapositive inference for inequality. (Contributed by NM, 9-Aug-2006.)

Theoremnecon3ai 2452 Contrapositive inference for inequality. (Contributed by NM, 23-May-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon3bi 2453 Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon1ai 2454 Contrapositive inference for inequality. (Contributed by NM, 12-Feb-2007.)

Theoremnecon1bi 2455 Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon1i 2456 Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.)

Theoremnecon2ai 2457 Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon2bi 2458 Contrapositive inference for inequality. (Contributed by NM, 1-Apr-2007.)

Theoremnecon2i 2459 Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.)

Theoremnecon2ad 2460 Contrapositive inference for inequality. (Contributed by NM, 19-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon2bd 2461 Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)

Theoremnecon2d 2462 Contrapositive inference for inequality. (Contributed by NM, 28-Dec-2008.)

Theoremnecon1abii 2463 Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.)

Theoremnecon1bbii 2464 Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.)

Theoremnecon1abid 2465 Contrapositive deduction for inequality. (Contributed by NM, 21-Aug-2007.)

Theoremnecon1bbid 2466 Contrapositive inference for inequality. (Contributed by NM, 31-Jan-2008.)

Theoremnecon2abii 2467 Contrapositive inference for inequality. (Contributed by NM, 2-Mar-2007.)

Theoremnecon2bbii 2468 Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)

Theoremnecon2abid 2469 Contrapositive deduction for inequality. (Contributed by NM, 18-Jul-2007.)

Theoremnecon2bbid 2470 Contrapositive deduction for inequality. (Contributed by NM, 13-Apr-2007.)

Theoremnecon4ai 2471 Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon4i 2472 Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon4ad 2473 Contrapositive inference for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon4bd 2474 Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon4d 2475 Contrapositive inference for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon4abid 2476 Contrapositive law deduction for inequality. (Contributed by NM, 11-Jan-2008.)

Theoremnecon4bbid 2477 Contrapositive law deduction for inequality. (Contributed by NM, 9-May-2012.)

Theoremnecon4bid 2478 Contrapositive law deduction for inequality. (Contributed by NM, 29-Jun-2007.)

Theoremnecon1ad 2479 Contrapositive deduction for inequality. (Contributed by NM, 2-Apr-2007.)

Theoremnecon1bd 2480 Contrapositive deduction for inequality. (Contributed by NM, 21-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon1d 2481 Contrapositive law deduction for inequality. (Contributed by NM, 28-Dec-2008.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremneneqad 2482 If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2428. One-way deduction form of df-ne 2414. (Contributed by David Moews, 28-Feb-2017.)

Theoremnebi 2483 Contraposition law for inequality. (Contributed by NM, 28-Dec-2008.)

Theorempm13.18 2484 Theorem *13.18 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)

Theorempm13.181 2485 Theorem *13.181 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)

Theorempm2.21ddne 2486 A contradiction implies anything. Equality/inequality deduction form. (Contributed by David Moews, 28-Feb-2017.)

Theorempm2.61ne 2487 Deduction eliminating an inequality in an antecedent. (Contributed by NM, 24-May-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theorempm2.61ine 2488 Inference eliminating an inequality in an antecedent. (Contributed by NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theorempm2.61dne 2489 Deduction eliminating an inequality in an antecedent. (Contributed by NM, 1-Jun-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theorempm2.61dane 2490 Deduction eliminating an inequality in an antecedent. (Contributed by NM, 30-Nov-2011.)

Theorempm2.61da2ne 2491 Deduction eliminating two inequalities in an antecedent. (Contributed by NM, 29-May-2013.)

Theorempm2.61da3ne 2492 Deduction eliminating three inequalities in an antecedent. (Contributed by NM, 15-Jun-2013.)

Theoremnecom 2493 Commutation of inequality. (Contributed by NM, 14-May-1999.)

Theoremnecomi 2494 Inference from commutative law for inequality. (Contributed by NM, 17-Oct-2012.)

Theoremnecomd 2495 Deduction from commutative law for inequality. (Contributed by NM, 12-Feb-2008.)

Theoremneor 2496 Logical OR with an equality. (Contributed by NM, 29-Apr-2007.)

Theoremneanior 2497 A DeMorgan's law for inequality. (Contributed by NM, 18-May-2007.)

Theoremne3anior 2498 A DeMorgan's law for inequality. (Contributed by NM, 30-Sep-2013.)

Theoremneorian 2499 A DeMorgan's law for inequality. (Contributed by NM, 18-May-2007.)

Theoremnemtbir 2500 An inference from an inequality, related to modus tollens. (Contributed by NM, 13-Apr-2007.)

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