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Type | Label | Description |
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Theorem | difrab0eqim 3501* | If the difference between the restricting class of a restricted class abstraction and the restricted class abstraction is empty, the restricting class is equal to this restricted class abstraction. (Contributed by Jim Kingdon, 3-Aug-2018.) |
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Theorem | inssdif0im 3502 | Intersection, subclass, and difference relationship. In classical logic the converse would also hold. (Contributed by Jim Kingdon, 3-Aug-2018.) |
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Theorem | difid 3503 | The difference between a class and itself is the empty set. Proposition 5.15 of [TakeutiZaring] p. 20. Also Theorem 32 of [Suppes] p. 28. (Contributed by NM, 22-Apr-2004.) |
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Theorem | difidALT 3504 | The difference between a class and itself is the empty set. Proposition 5.15 of [TakeutiZaring] p. 20. Also Theorem 32 of [Suppes] p. 28. Alternate proof of difid 3503. (Contributed by David Abernethy, 17-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.) |
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Theorem | dif0 3505 | The difference between a class and the empty set. Part of Exercise 4.4 of [Stoll] p. 16. (Contributed by NM, 17-Aug-2004.) |
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Theorem | 0dif 3506 | The difference between the empty set and a class. Part of Exercise 4.4 of [Stoll] p. 16. (Contributed by NM, 17-Aug-2004.) |
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Theorem | disjdif 3507 | A class and its relative complement are disjoint. Theorem 38 of [Suppes] p. 29. (Contributed by NM, 24-Mar-1998.) |
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Theorem | difin0 3508 | The difference of a class from its intersection is empty. Theorem 37 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
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Theorem | undif1ss 3509 | Absorption of difference by union. In classical logic, as Theorem 35 of [Suppes] p. 29, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.) |
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Theorem | undif2ss 3510 | Absorption of difference by union. In classical logic, as in Part of proof of Corollary 6K of [Enderton] p. 144, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.) |
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Theorem | undifabs 3511 | Absorption of difference by union. (Contributed by NM, 18-Aug-2013.) |
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Theorem | inundifss 3512 | The intersection and class difference of a class with another class are contained in the original class. In classical logic we'd be able to make a stronger statement: that everything in the original class is in the intersection or the difference (that is, this theorem would be equality rather than subset). (Contributed by Jim Kingdon, 4-Aug-2018.) |
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Theorem | disjdif2 3513 | The difference of a class and a class disjoint from it is the original class. (Contributed by BJ, 21-Apr-2019.) |
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Theorem | difun2 3514 | Absorption of union by difference. Theorem 36 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.) |
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Theorem | undifss 3515 | Union of complementary parts into whole. (Contributed by Jim Kingdon, 4-Aug-2018.) |
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Theorem | ssdifin0 3516 | A subset of a difference does not intersect the subtrahend. (Contributed by Jeff Hankins, 1-Sep-2013.) (Proof shortened by Mario Carneiro, 24-Aug-2015.) |
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Theorem | ssdifeq0 3517 | A class is a subclass of itself subtracted from another iff it is the empty set. (Contributed by Steve Rodriguez, 20-Nov-2015.) |
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Theorem | ssundifim 3518 | A consequence of inclusion in the union of two classes. In classical logic this would be a biconditional. (Contributed by Jim Kingdon, 4-Aug-2018.) |
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Theorem | difdifdirss 3519 | Distributive law for class difference. In classical logic, as in Exercise 4.8 of [Stoll] p. 16, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.) |
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Theorem | uneqdifeqim 3520 |
Two ways that ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | r19.2m 3521* | Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1648). The restricted version is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.) (Revised by Jim Kingdon, 7-Apr-2023.) |
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Theorem | r19.2mOLD 3522* | Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1648). The restricted version is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.) Obsolete version of r19.2m 3521 as of 7-Apr-2023. (Proof modification is discouraged.) (New usage is discouraged.) |
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Theorem | r19.3rm 3523* | Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 19-Dec-2018.) |
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Theorem | r19.28m 3524* | Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.) |
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Theorem | r19.3rmv 3525* | Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 6-Aug-2018.) |
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Theorem | r19.9rmv 3526* | Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 5-Aug-2018.) |
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Theorem | r19.28mv 3527* | Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 6-Aug-2018.) |
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Theorem | r19.45mv 3528* | Restricted version of Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.) |
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Theorem | r19.44mv 3529* | Restricted version of Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.) |
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Theorem | r19.27m 3530* | Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.) |
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Theorem | r19.27mv 3531* | Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.) |
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Theorem | rzal 3532* | Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
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Theorem | rexn0 3533* | Restricted existential quantification implies its restriction is nonempty (it is also inhabited as shown in rexm 3534). (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) |
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Theorem | rexm 3534* | Restricted existential quantification implies its restriction is inhabited. (Contributed by Jim Kingdon, 16-Oct-2018.) |
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Theorem | ralidm 3535* | Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997.) |
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Theorem | ral0 3536 | Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.) |
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Theorem | rgenm 3537* | Generalization rule that eliminates an inhabited class requirement. (Contributed by Jim Kingdon, 5-Aug-2018.) |
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Theorem | ralf0 3538* | The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.) |
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Theorem | ralm 3539 | Inhabited classes and restricted quantification. (Contributed by Jim Kingdon, 6-Aug-2018.) |
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Theorem | raaanlem 3540* |
Special case of raaan 3541 where ![]() |
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Theorem | raaan 3541* | Rearrange restricted quantifiers. (Contributed by NM, 26-Oct-2010.) |
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Theorem | raaanv 3542* | Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.) |
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Theorem | sbss 3543* | Set substitution into the first argument of a subset relation. (Contributed by Rodolfo Medina, 7-Jul-2010.) (Proof shortened by Mario Carneiro, 14-Nov-2016.) |
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Theorem | sbcssg 3544 | Distribute proper substitution through a subclass relation. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Alexander van der Vekens, 23-Jul-2017.) |
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Theorem | dcun 3545 | The union of two decidable classes is decidable. (Contributed by Jim Kingdon, 5-Oct-2022.) |
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Syntax | cif 3546 | Extend class notation to include the conditional operator. See df-if 3547 for a description. (In older databases this was denoted "ded".) |
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Definition | df-if 3547* |
Define the conditional operator. Read ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]()
In the absence of excluded middle, this will tend to be useful where
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Theorem | dfif6 3548* | An alternate definition of the conditional operator df-if 3547 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.) |
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Theorem | ifeq1 3549 | Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
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Theorem | ifeq2 3550 | Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
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Theorem | iftrue 3551 | Value of the conditional operator when its first argument is true. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
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Theorem | iftruei 3552 | Inference associated with iftrue 3551. (Contributed by BJ, 7-Oct-2018.) |
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Theorem | iftrued 3553 | Value of the conditional operator when its first argument is true. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | iffalse 3554 | Value of the conditional operator when its first argument is false. (Contributed by NM, 14-Aug-1999.) |
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Theorem | iffalsei 3555 | Inference associated with iffalse 3554. (Contributed by BJ, 7-Oct-2018.) |
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Theorem | iffalsed 3556 | Value of the conditional operator when its first argument is false. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | ifnefalse 3557 | When values are unequal, but an "if" condition checks if they are equal, then the "false" branch results. This is a simple utility to provide a slight shortening and simplification of proofs versus applying iffalse 3554 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.) |
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Theorem | ifsbdc 3558 | Distribute a function over an if-clause. (Contributed by Jim Kingdon, 1-Jan-2022.) |
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Theorem | dfif3 3559* |
Alternate definition of the conditional operator df-if 3547. Note that
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Theorem | ifssun 3560 | A conditional class is included in the union of its two alternatives. (Contributed by BJ, 15-Aug-2024.) |
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Theorem | ifidss 3561 | A conditional class whose two alternatives are equal is included in that alternative. With excluded middle, we can prove it is equal to it. (Contributed by BJ, 15-Aug-2024.) |
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Theorem | ifeq12 3562 | Equality theorem for conditional operators. (Contributed by NM, 1-Sep-2004.) |
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Theorem | ifeq1d 3563 | Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.) |
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Theorem | ifeq2d 3564 | Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.) |
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Theorem | ifeq12d 3565 | Equality deduction for conditional operator. (Contributed by NM, 24-Mar-2015.) |
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Theorem | ifbi 3566 | Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.) |
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Theorem | ifbid 3567 | Equivalence deduction for conditional operators. (Contributed by NM, 18-Apr-2005.) |
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Theorem | ifbieq1d 3568 | Equivalence/equality deduction for conditional operators. (Contributed by JJ, 25-Sep-2018.) |
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Theorem | ifbieq2i 3569 | Equivalence/equality inference for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.) |
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Theorem | ifbieq2d 3570 | Equivalence/equality deduction for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.) |
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Theorem | ifbieq12i 3571 | Equivalence deduction for conditional operators. (Contributed by NM, 18-Mar-2013.) |
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Theorem | ifbieq12d 3572 | Equivalence deduction for conditional operators. (Contributed by Jeff Madsen, 2-Sep-2009.) |
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Theorem | nfifd 3573 | Deduction version of nfif 3574. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 13-Oct-2016.) |
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Theorem | nfif 3574 | Bound-variable hypothesis builder for a conditional operator. (Contributed by NM, 16-Feb-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
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Theorem | ifcldadc 3575 | Conditional closure. (Contributed by Jim Kingdon, 11-Jan-2022.) |
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Theorem | ifeq1dadc 3576 | Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.) |
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Theorem | ifeq2dadc 3577 | Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.) |
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Theorem | ifbothdadc 3578 |
A formula ![]() |
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Theorem | ifbothdc 3579 |
A wff ![]() |
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Theorem | ifiddc 3580 | Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.) |
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Theorem | eqifdc 3581 | Expansion of an equality with a conditional operator. (Contributed by Jim Kingdon, 28-Jul-2022.) |
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Theorem | ifcldcd 3582 | Membership (closure) of a conditional operator, deduction form. (Contributed by Jim Kingdon, 8-Aug-2021.) |
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Theorem | ifnotdc 3583 | Negating the first argument swaps the last two arguments of a conditional operator. (Contributed by NM, 21-Jun-2007.) |
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Theorem | ifandc 3584 | Rewrite a conjunction in a conditional as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.) |
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Theorem | ifordc 3585 | Rewrite a disjunction in a conditional as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.) |
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Theorem | ifmdc 3586 | If a conditional class is inhabited, then the condition is decidable. This shows that conditionals are not very useful unless one can prove the condition decidable. (Contributed by BJ, 24-Sep-2022.) |
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Syntax | cpw 3587 | Extend class notation to include power class. (The tilde in the Metamath token is meant to suggest the calligraphic font of the P.) |
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Theorem | pwjust 3588* | Soundness justification theorem for df-pw 3589. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
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Definition | df-pw 3589* |
Define power class. Definition 5.10 of [TakeutiZaring] p. 17, but we
also let it apply to proper classes, i.e. those that are not members of
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Theorem | pweq 3590 | Equality theorem for power class. (Contributed by NM, 5-Aug-1993.) |
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Theorem | pweqi 3591 | Equality inference for power class. (Contributed by NM, 27-Nov-2013.) |
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Theorem | pweqd 3592 | Equality deduction for power class. (Contributed by NM, 27-Nov-2013.) |
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Theorem | elpw 3593 | Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 31-Dec-1993.) |
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Theorem | velpw 3594* | Setvar variable membership in a power class (common case). See elpw 3593. (Contributed by David A. Wheeler, 8-Dec-2018.) |
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Theorem | elpwg 3595 | Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 6-Aug-2000.) |
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Theorem | elpwi 3596 | Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.) |
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Theorem | elpwb 3597 | Characterization of the elements of a power class. (Contributed by BJ, 29-Apr-2021.) |
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Theorem | elpwid 3598 | An element of a power class is a subclass. Deduction form of elpwi 3596. (Contributed by David Moews, 1-May-2017.) |
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Theorem | elelpwi 3599 |
If ![]() ![]() ![]() ![]() |
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Theorem | nfpw 3600 | Bound-variable hypothesis builder for power class. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) |
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