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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.)
 F/_   =>    
 F/
 
Theoremnfcrd 2502* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
 F/_   =>     F/
 
Theoremnfeqd 2503 Hypothesis builder for equality. (Contributed by Mario Carneiro, 7-Oct-2016.)
 F/_   &     F/_   =>     F/
 
Theoremnfeld 2504 Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 7-Oct-2016.)
 F/_   &     F/_   =>     F/
 
Theoremdrnfc1 2505 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 8-Oct-2016.)
   =>     F/_  F/_
 
Theoremdrnfc2 2506 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 8-Oct-2016.)
   =>     F/_  F/_
 
Theoremnfabd2 2507 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 8-Oct-2016.)

 F/   &     F/   =>     F/_
 
Theoremnfabd 2508 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 8-Oct-2016.)

 F/   &     F/   =>     F/_
 
Theoremdvelimdc 2509 Deduction form of dvelimc 2510. (Contributed by Mario Carneiro, 8-Oct-2016.)

 F/   &     F/   &     F/_   &     F/_   &       =>     F/_
 
Theoremdvelimc 2510 Version of dvelim 2016 for classes. (Contributed by Mario Carneiro, 8-Oct-2016.)
 F/_   &     F/_   &       =>     F/_
 
Theoremnfcvf 2511 If and are distinct, then is not free in . (Contributed by Mario Carneiro, 8-Oct-2016.)
 F/_
 
Theoremnfcvf2 2512 If and are distinct, then is not free in . (Contributed by Mario Carneiro, 5-Dec-2016.)
 F/_
 
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.)
 F/_   &     F/_   =>   
 
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.)
 F/_   =>   
 
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.)
 F/_   =>   
 
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.)
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