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Theorem List for Metamath Proof Explorer - 3401-3500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempsseq12d 3401 An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)

Theorempssss 3402 A proper subclass is a subclass. Theorem 10 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)

Theorempssne 3403 Two classes in a proper subclass relationship are not equal. (Contributed by NM, 16-Feb-2015.)

Theorempssssd 3404 Deduce subclass from proper subclass. (Contributed by NM, 29-Feb-1996.)

Theorempssned 3405 Proper subclasses are unequal. Deduction form of pssne 3403. (Contributed by David Moews, 1-May-2017.)

Theoremsspss 3406 Subclass in terms of proper subclass. (Contributed by NM, 25-Feb-1996.)

Theorempssirr 3407 Proper subclass is irreflexive. Theorem 7 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)

Theorempssn2lp 3408 Proper subclass has no 2-cycle loops. Compare Theorem 8 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremsspsstri 3409 Two ways of stating trichotomy with respect to inclusion. (Contributed by NM, 12-Aug-2004.)

Theoremssnpss 3410 Partial trichotomy law for subclasses. (Contributed by NM, 16-May-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theorempsstr 3411 Transitive law for proper subclass. Theorem 9 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)

Theoremsspsstr 3412 Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)

Theorempsssstr 3413 Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)

Theorempsstrd 3414 Proper subclass inclusion is transitive. Deduction form of psstr 3411. (Contributed by David Moews, 1-May-2017.)

Theoremsspsstrd 3415 Transitivity involving subclass and proper subclass inclusion. Deduction form of sspsstr 3412. (Contributed by David Moews, 1-May-2017.)

Theorempsssstrd 3416 Transitivity involving subclass and proper subclass inclusion. Deduction form of psssstr 3413. (Contributed by David Moews, 1-May-2017.)

Theoremnpss 3417 A class is not a proper subclass of another iff it satisfies a one-directional form of eqss 3323. (Contributed by Mario Carneiro, 15-May-2015.)

2.1.13  The difference, union, and intersection of two classes

2.1.13.1  The difference of two classes

Theoremdifeq1 3418 Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdifeq2 3419 Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdifeq12 3420 Equality theorem for class difference. (Contributed by FL, 31-Aug-2009.)

Theoremdifeq1i 3421 Inference adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)

Theoremdifeq2i 3422 Inference adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)

Theoremdifeq12i 3423 Equality inference for class difference. (Contributed by NM, 29-Aug-2004.)

Theoremdifeq1d 3424 Deduction adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)

Theoremdifeq2d 3425 Deduction adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)

Theoremdifeq12d 3426 Equality deduction for class difference. (Contributed by FL, 29-May-2014.)

Theoremdifeqri 3427* Inference from membership to difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremnfdif 3428 Bound-variable hypothesis builder for class difference. (Contributed by NM, 3-Dec-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theoremeldifi 3429 Implication of membership in a class difference. (Contributed by NM, 29-Apr-1994.)

Theoremeldifn 3430 Implication of membership in a class difference. (Contributed by NM, 3-May-1994.)

Theoremelndif 3431 A set does not belong to a class excluding it. (Contributed by NM, 27-Jun-1994.)

Theoremneldif 3432 Implication of membership in a class difference. (Contributed by NM, 28-Jun-1994.)

Theoremdifdif 3433 Double class difference. Exercise 11 of [TakeutiZaring] p. 22. (Contributed by NM, 17-May-1998.)

Theoremdifss 3434 Subclass relationship for class difference. Exercise 14 of [TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.)

Theoremdifssd 3435 A difference of two classes is contained in the minuend. Deduction form of difss 3434. (Contributed by David Moews, 1-May-2017.)

Theoremdifss2 3436 If a class is contained in a difference, it is contained in the minuend. (Contributed by David Moews, 1-May-2017.)

Theoremdifss2d 3437 If a class is contained in a difference, it is contained in the minuend. Deduction form of difss2 3436. (Contributed by David Moews, 1-May-2017.)

Theoremssdifss 3438 Preservation of a subclass relationship by class difference. (Contributed by NM, 15-Feb-2007.)

Theoremddif 3439 Double complement under universal class. Exercise 4.10(s) of [Mendelson] p. 231. (Contributed by NM, 8-Jan-2002.)

Theoremssconb 3440 Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.)

Theoremsscon 3441 Contraposition law for subsets. Exercise 15 of [TakeutiZaring] p. 22. (Contributed by NM, 22-Mar-1998.)

Theoremssdif 3442 Difference law for subsets. (Contributed by NM, 28-May-1998.)

Theoremssdifd 3443 If is contained in , then is contained in . Deduction form of ssdif 3442. (Contributed by David Moews, 1-May-2017.)

Theoremsscond 3444 If is contained in , then is contained in . Deduction form of sscon 3441. (Contributed by David Moews, 1-May-2017.)

Theoremssdifssd 3445 If is contained in , then is also contained in . Deduction form of ssdifss 3438. (Contributed by David Moews, 1-May-2017.)

Theoremssdif2d 3446 If is contained in and is contained in , then is contained in . Deduction form. (Contributed by David Moews, 1-May-2017.)

Theoremraldifb 3447 Restricted universal quantification on a class difference in terms of an implication. (Contributed by Alexander van der Vekens, 3-Jan-2018.)

2.1.13.2  The union of two classes

Theoremelun 3448 Expansion of membership in class union. Theorem 12 of [Suppes] p. 25. (Contributed by NM, 7-Aug-1994.)

Theoremuneqri 3449* Inference from membership to union. (Contributed by NM, 5-Aug-1993.)

Theoremunidm 3450 Idempotent law for union of classes. Theorem 23 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)

Theoremuncom 3451 Commutative law for union of classes. Exercise 6 of [TakeutiZaring] p. 17. (Contributed by NM, 25-Jun-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremequncom 3452 If a class equals the union of two other classes, then it equals the union of those two classes commuted. equncom 3452 was automatically derived from equncomVD 28689 using the tools program translatewithout_overwriting.cmd and minimizing. (Contributed by Alan Sare, 18-Feb-2012.)

Theoremequncomi 3453 Inference form of equncom 3452. equncomi 3453 was automatically derived from equncomiVD 28690 using the tools program translatewithout_overwriting.cmd and minimizing. (Contributed by Alan Sare, 18-Feb-2012.)

Theoremuneq1 3454 Equality theorem for union of two classes. (Contributed by NM, 5-Aug-1993.)

Theoremuneq2 3455 Equality theorem for the union of two classes. (Contributed by NM, 5-Aug-1993.)

Theoremuneq12 3456 Equality theorem for union of two classes. (Contributed by NM, 29-Mar-1998.)

Theoremuneq1i 3457 Inference adding union to the right in a class equality. (Contributed by NM, 30-Aug-1993.)

Theoremuneq2i 3458 Inference adding union to the left in a class equality. (Contributed by NM, 30-Aug-1993.)

Theoremuneq12i 3459 Equality inference for union of two classes. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Theoremuneq1d 3460 Deduction adding union to the right in a class equality. (Contributed by NM, 29-Mar-1998.)

Theoremuneq2d 3461 Deduction adding union to the left in a class equality. (Contributed by NM, 29-Mar-1998.)

Theoremuneq12d 3462 Equality deduction for union of two classes. (Contributed by NM, 29-Sep-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremnfun 3463 Bound-variable hypothesis builder for the union of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.)

Theoremunass 3464 Associative law for union of classes. Exercise 8 of [TakeutiZaring] p. 17. (Contributed by NM, 3-May-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremun12 3465 A rearrangement of union. (Contributed by NM, 12-Aug-2004.)

Theoremun23 3466 A rearrangement of union. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremun4 3467 A rearrangement of the union of 4 classes. (Contributed by NM, 12-Aug-2004.)

Theoremunundi 3468 Union distributes over itself. (Contributed by NM, 17-Aug-2004.)

Theoremunundir 3469 Union distributes over itself. (Contributed by NM, 17-Aug-2004.)

Theoremssun1 3470 Subclass relationship for union of classes. Theorem 25 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)

Theoremssun2 3471 Subclass relationship for union of classes. (Contributed by NM, 30-Aug-1993.)

Theoremssun3 3472 Subclass law for union of classes. (Contributed by NM, 5-Aug-1993.)

Theoremssun4 3473 Subclass law for union of classes. (Contributed by NM, 14-Aug-1994.)

Theoremelun1 3474 Membership law for union of classes. (Contributed by NM, 5-Aug-1993.)

Theoremelun2 3475 Membership law for union of classes. (Contributed by NM, 30-Aug-1993.)

Theoremunss1 3476 Subclass law for union of classes. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremssequn1 3477 A relationship between subclass and union. Theorem 26 of [Suppes] p. 27. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremunss2 3478 Subclass law for union of classes. Exercise 7 of [TakeutiZaring] p. 18. (Contributed by NM, 14-Oct-1999.)

Theoremunss12 3479 Subclass law for union of classes. (Contributed by NM, 2-Jun-2004.)

Theoremssequn2 3480 A relationship between subclass and union. (Contributed by NM, 13-Jun-1994.)

Theoremunss 3481 The union of two subclasses is a subclass. Theorem 27 of [Suppes] p. 27 and its converse. (Contributed by NM, 11-Jun-2004.)

Theoremunssi 3482 An inference showing the union of two subclasses is a subclass. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremunssd 3483 A deduction showing the union of two subclasses is a subclass. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremunssad 3484 If is contained in , so is . One-way deduction form of unss 3481. Partial converse of unssd 3483. (Contributed by David Moews, 1-May-2017.)

Theoremunssbd 3485 If is contained in , so is . One-way deduction form of unss 3481. Partial converse of unssd 3483. (Contributed by David Moews, 1-May-2017.)

Theoremssun 3486 A condition that implies inclusion in the union of two classes. (Contributed by NM, 23-Nov-2003.)

Theoremrexun 3487 Restricted existential quantification over union. (Contributed by Jeff Madsen, 5-Jan-2011.)

Theoremralunb 3488 Restricted quantification over a union. (Contributed by Scott Fenton, 12-Apr-2011.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremralun 3489 Restricted quantification over union. (Contributed by Jeff Madsen, 2-Sep-2009.)

2.1.13.3  The intersection of two classes

Theoremelin 3490 Expansion of membership in an intersection of two classes. Theorem 12 of [Suppes] p. 25. (Contributed by NM, 29-Apr-1994.)

Theoremelin2 3491 Membership in a class defined as an intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)

Theoremelin3 3492 Membership in a class defined as a ternary intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)

Theoremincom 3493 Commutative law for intersection of classes. Exercise 7 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.)

Theoremineqri 3494* Inference from membership to intersection. (Contributed by NM, 5-Aug-1993.)

Theoremineq1 3495 Equality theorem for intersection of two classes. (Contributed by NM, 14-Dec-1993.)

Theoremineq2 3496 Equality theorem for intersection of two classes. (Contributed by NM, 26-Dec-1993.)

Theoremineq12 3497 Equality theorem for intersection of two classes. (Contributed by NM, 8-May-1994.)

Theoremineq1i 3498 Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.)

Theoremineq2i 3499 Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.)

Theoremineq12i 3500 Equality inference for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

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