Home New Foundations ExplorerTheorem List (p. 26 of 64) < Previous  Next > Browser slow? Try the Unicode version. Mirrors  >  Metamath Home Page  >  NFE Home Page  >  Theorem List Contents       This page: Page List

Theorem List for New Foundations Explorer - 2501-2600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnfel2 2501* Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)

Theoremnfcrd 2502* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfeqd 2503 Hypothesis builder for equality. (Contributed by Mario Carneiro, 7-Oct-2016.)

Theoremnfeld 2504 Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 7-Oct-2016.)

Theoremdrnfc1 2505 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 8-Oct-2016.)

Theoremdrnfc2 2506 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 8-Oct-2016.)

Theoremnfabd2 2507 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 8-Oct-2016.)

Theoremnfabd 2508 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 8-Oct-2016.)

Theoremdvelimdc 2509 Deduction form of dvelimc 2510. (Contributed by Mario Carneiro, 8-Oct-2016.)

Theoremdvelimc 2510 Version of dvelim 2016 for classes. (Contributed by Mario Carneiro, 8-Oct-2016.)

Theoremnfcvf 2511 If and are distinct, then is not free in . (Contributed by Mario Carneiro, 8-Oct-2016.)

Theoremnfcvf2 2512 If and are distinct, then is not free in . (Contributed by Mario Carneiro, 5-Dec-2016.)

Theoremcleqf 2513 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 2514 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 2515* 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 2516 Extend wff notation to include inequality.

Syntaxwnel 2517 Extend wff notation to include negated membership.

Definitiondf-ne 2518 Define inequality. (Contributed by NM, 5-Aug-1993.)

Definitiondf-nel 2519 Define negated membership. (Contributed by NM, 7-Aug-1994.)

Theoremnne 2520 Negation of inequality. (Contributed by NM, 9-Jun-2006.)

Theoremneirr 2521 No class is unequal to itself. (Contributed by Stefan O'Rear, 1-Jan-2015.)

Theoremexmidne 2522 Excluded middle with equality and inequality. (Contributed by NM, 3-Feb-2012.)

Theoremnonconne 2523 Law of noncontradiction with equality and inequality. (Contributed by NM, 3-Feb-2012.)

Theoremneeq1 2524 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)

Theoremneeq2 2525 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)

Theoremneeq1i 2526 Inference for inequality. (Contributed by NM, 29-Apr-2005.)

Theoremneeq2i 2527 Inference for inequality. (Contributed by NM, 29-Apr-2005.)

Theoremneeq12i 2528 Inference for inequality. (Contributed by NM, 24-Jul-2012.)

Theoremneeq1d 2529 Deduction for inequality. (Contributed by NM, 25-Oct-1999.)

Theoremneeq2d 2530 Deduction for inequality. (Contributed by NM, 25-Oct-1999.)

Theoremneeq12d 2531 Deduction for inequality. (Contributed by NM, 24-Jul-2012.)

Theoremneneqd 2532 Deduction eliminating inequality definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremeqnetri 2533 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Theoremeqnetrd 2534 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Theoremeqnetrri 2535 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Theoremeqnetrrd 2536 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Theoremneeqtri 2537 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Theoremneeqtrd 2538 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Theoremneeqtrri 2539 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Theoremneeqtrrd 2540 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Theoremsyl5eqner 2541 B chained equality inference for inequality. (Contributed by NM, 6-Jun-2012.)

Theorem3netr3d 2542 Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)

Theorem3netr4d 2543 Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)

Theorem3netr3g 2544 Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)

Theorem3netr4g 2545 Substitution of equality into both sides of an inequality. (Contributed by NM, 14-Jun-2012.)

Theoremnecon3abii 2546 Deduction from equality to inequality. (Contributed by NM, 9-Nov-2007.)

Theoremnecon3bbii 2547 Deduction from equality to inequality. (Contributed by NM, 13-Apr-2007.)

Theoremnecon3bii 2548 Inference from equality to inequality. (Contributed by NM, 23-Feb-2005.)

Theoremnecon3abid 2549 Deduction from equality to inequality. (Contributed by NM, 21-Mar-2007.)

Theoremnecon3bbid 2550 Deduction from equality to inequality. (Contributed by NM, 2-Jun-2007.)

Theoremnecon3bid 2551 Deduction from equality to inequality. (Contributed by NM, 23-Feb-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon3ad 2552 Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon3bd 2553 Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon3d 2554 Contrapositive law deduction for inequality. (Contributed by NM, 10-Jun-2006.)

Theoremnecon3i 2555 Contrapositive inference for inequality. (Contributed by NM, 9-Aug-2006.)

Theoremnecon3ai 2556 Contrapositive inference for inequality. (Contributed by NM, 23-May-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon3bi 2557 Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon1ai 2558 Contrapositive inference for inequality. (Contributed by NM, 12-Feb-2007.)

Theoremnecon1bi 2559 Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon1i 2560 Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.)

Theoremnecon2ai 2561 Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon2bi 2562 Contrapositive inference for inequality. (Contributed by NM, 1-Apr-2007.)

Theoremnecon2i 2563 Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.)

Theoremnecon2ad 2564 Contrapositive inference for inequality. (Contributed by NM, 19-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon2bd 2565 Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)

Theoremnecon2d 2566 Contrapositive inference for inequality. (Contributed by NM, 28-Dec-2008.)

Theoremnecon1abii 2567 Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.)

Theoremnecon1bbii 2568 Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.)

Theoremnecon1abid 2569 Contrapositive deduction for inequality. (Contributed by NM, 21-Aug-2007.)

Theoremnecon1bbid 2570 Contrapositive inference for inequality. (Contributed by NM, 31-Jan-2008.)

Theoremnecon2abii 2571 Contrapositive inference for inequality. (Contributed by NM, 2-Mar-2007.)

Theoremnecon2bbii 2572 Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)

Theoremnecon2abid 2573 Contrapositive deduction for inequality. (Contributed by NM, 18-Jul-2007.)

Theoremnecon2bbid 2574 Contrapositive deduction for inequality. (Contributed by NM, 13-Apr-2007.)

Theoremnecon4ai 2575 Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon4i 2576 Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon4ad 2577 Contrapositive inference for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon4bd 2578 Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon4d 2579 Contrapositive inference for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon4abid 2580 Contrapositive law deduction for inequality. (Contributed by NM, 11-Jan-2008.)

Theoremnecon4bbid 2581 Contrapositive law deduction for inequality. (Contributed by NM, 9-May-2012.)

Theoremnecon4bid 2582 Contrapositive law deduction for inequality. (Contributed by NM, 29-Jun-2007.)

Theoremnecon1ad 2583 Contrapositive deduction for inequality. (Contributed by NM, 2-Apr-2007.)

Theoremnecon1bd 2584 Contrapositive deduction for inequality. (Contributed by NM, 21-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnecon1d 2585 Contrapositive law deduction for inequality. (Contributed by NM, 28-Dec-2008.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremneneqad 2586 If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2532. One-way deduction form of df-ne 2518. (Contributed by David Moews, 28-Feb-2017.)

Theoremnebi 2587 Contraposition law for inequality. (Contributed by NM, 28-Dec-2008.)

Theorempm13.18 2588 Theorem *13.18 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)

Theorempm13.181 2589 Theorem *13.181 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)

Theorempm2.21ddne 2590 A contradiction implies anything. Equality/inequality deduction form. (Contributed by David Moews, 28-Feb-2017.)

Theorempm2.61ne 2591 Deduction eliminating an inequality in an antecedent. (Contributed by NM, 24-May-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theorempm2.61ine 2592 Inference eliminating an inequality in an antecedent. (Contributed by NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theorempm2.61dne 2593 Deduction eliminating an inequality in an antecedent. (Contributed by NM, 1-Jun-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theorempm2.61dane 2594 Deduction eliminating an inequality in an antecedent. (Contributed by NM, 30-Nov-2011.)

Theorempm2.61da2ne 2595 Deduction eliminating two inequalities in an antecedent. (Contributed by NM, 29-May-2013.)

Theorempm2.61da3ne 2596 Deduction eliminating three inequalities in an antecedent. (Contributed by NM, 15-Jun-2013.)

Theoremnecom 2597 Commutation of inequality. (Contributed by NM, 14-May-1999.)

Theoremnecomi 2598 Inference from commutative law for inequality. (Contributed by NM, 17-Oct-2012.)

Theoremnecomd 2599 Deduction from commutative law for inequality. (Contributed by NM, 12-Feb-2008.)

Theoremneor 2600 Logical OR with an equality. (Contributed by NM, 29-Apr-2007.)

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6337
 Copyright terms: Public domain < Previous  Next >