Home | Metamath
Proof Explorer Theorem List (p. 45 of 426) | < Previous Next > |
Browser slow? Try the
Unicode version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | Metamath Proof Explorer
(1-27775) |
Hilbert Space Explorer
(27776-29300) |
Users' Mathboxes
(29301-42551) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | dfop 4401 | Value of an ordered pair when the arguments are sets, with the conclusion corresponding to Kuratowski's original definition. (Contributed by NM, 25-Jun-1998.) (Avoid depending on this detail.) |
Theorem | opeq1 4402 | Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Theorem | opeq2 4403 | Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Theorem | opeq12 4404 | Equality theorem for ordered pairs. (Contributed by NM, 28-May-1995.) |
Theorem | opeq1i 4405 | Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.) |
Theorem | opeq2i 4406 | Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.) |
Theorem | opeq12i 4407 | Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.) (Proof shortened by Eric Schmidt, 4-Apr-2007.) |
Theorem | opeq1d 4408 | Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.) |
Theorem | opeq2d 4409 | Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.) |
Theorem | opeq12d 4410 | Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Theorem | oteq1 4411 | Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.) |
Theorem | oteq2 4412 | Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.) |
Theorem | oteq3 4413 | Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.) |
Theorem | oteq1d 4414 | Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Theorem | oteq2d 4415 | Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Theorem | oteq3d 4416 | Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Theorem | oteq123d 4417 | Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Theorem | nfop 4418 | Bound-variable hypothesis builder for ordered pairs. (Contributed by NM, 14-Nov-1995.) |
Theorem | nfopd 4419 | Deduction version of bound-variable hypothesis builder nfop 4418. This shows how the deduction version of a not-free theorem such as nfop 4418 can be created from the corresponding not-free inference theorem. (Contributed by NM, 4-Feb-2008.) |
Theorem | csbopg 4420 | Distribution of class substitution over ordered pairs. (Contributed by Drahflow, 25-Sep-2015.) (Revised by Mario Carneiro, 29-Oct-2015.) (Revised by ML, 25-Oct-2020.) |
Theorem | opid 4421 | The ordered pair in Kuratowski's representation. (Contributed by FL, 28-Dec-2011.) (Avoid depending on this detail.) |
Theorem | ralunsn 4422* | Restricted quantification over the union of a set and a singleton, using implicit substitution. (Contributed by Paul Chapman, 17-Nov-2012.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Theorem | 2ralunsn 4423* | Double restricted quantification over the union of a set and a singleton, using implicit substitution. (Contributed by Paul Chapman, 17-Nov-2012.) |
Theorem | opprc 4424 | Expansion of an ordered pair when either member is a proper class. (Contributed by Mario Carneiro, 26-Apr-2015.) |
Theorem | opprc1 4425 | Expansion of an ordered pair when the first member is a proper class. See also opprc 4424. (Contributed by NM, 10-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Theorem | opprc2 4426 | Expansion of an ordered pair when the second member is a proper class. See also opprc 4424. (Contributed by NM, 15-Nov-1994.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Theorem | oprcl 4427 | If an ordered pair has an element, then its arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.) |
Theorem | pwsn 4428 | The power set of a singleton. (Contributed by NM, 5-Jun-2006.) |
Theorem | pwsnALT 4429 | Alternate proof of pwsn 4428, more direct. (Contributed by NM, 5-Jun-2006.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | pwpr 4430 | The power set of an unordered pair. (Contributed by NM, 1-May-2009.) |
Theorem | pwtp 4431 | The power set of an unordered triple. (Contributed by Mario Carneiro, 2-Jul-2016.) |
Theorem | pwpwpw0 4432 | Compute the power set of the power set of the power set of the empty set. (See also pw0 4343 and pwpw0 4344.) (Contributed by NM, 2-May-2009.) |
Theorem | pwv 4433 | The power class of the universe is the universe. Exercise 4.12(d) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.) |
Theorem | prproe 4434* | For an element of a proper unordered pair of elements of a class , there is another (different) element of the class which is an element of the proper pair. (Contributed by AV, 18-Dec-2021.) |
Theorem | 3elpr2eq 4435 | If there are three elements in a proper unordered pair, and two of them are different from the third one, the two must be equal. (Contributed by AV, 19-Dec-2021.) |
Syntax | cuni 4436 | Extend class notation to include the union of a class (read: 'union ') |
Definition | df-uni 4437* | Define the union of a class i.e. the collection of all members of the members of the class. Definition 5.5 of [TakeutiZaring] p. 16. For example, (ex-uni 27283). This is similar to the union of two classes df-un 3579. (Contributed by NM, 23-Aug-1993.) |
Theorem | dfuni2 4438* | Alternate definition of class union. (Contributed by NM, 28-Jun-1998.) |
Theorem | eluni 4439* | Membership in class union. (Contributed by NM, 22-May-1994.) |
Theorem | eluni2 4440* | Membership in class union. Restricted quantifier version. (Contributed by NM, 31-Aug-1999.) |
Theorem | elunii 4441 | Membership in class union. (Contributed by NM, 24-Mar-1995.) |
Theorem | nfuni 4442 | Bound-variable hypothesis builder for union. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | nfunid 4443 | Deduction version of nfuni 4442. (Contributed by NM, 18-Feb-2013.) |
Theorem | unieq 4444 | Equality theorem for class union. Exercise 15 of [TakeutiZaring] p. 18. (Contributed by NM, 10-Aug-1993.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Theorem | unieqi 4445 | Inference of equality of two class unions. (Contributed by NM, 30-Aug-1993.) |
Theorem | unieqd 4446 | Deduction of equality of two class unions. (Contributed by NM, 21-Apr-1995.) |
Theorem | eluniab 4447* | Membership in union of a class abstraction. (Contributed by NM, 11-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2016.) |
Theorem | elunirab 4448* | Membership in union of a class abstraction. (Contributed by NM, 4-Oct-2006.) |
Theorem | unipr 4449 | The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 23-Aug-1993.) |
Theorem | uniprg 4450 | The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 25-Aug-2006.) |
Theorem | unisn 4451 | A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. (Contributed by NM, 30-Aug-1993.) |
Theorem | unisng 4452 | A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. (Contributed by NM, 13-Aug-2002.) |
Theorem | unisn3 4453* | Union of a singleton in the form of a restricted class abstraction. (Contributed by NM, 3-Jul-2008.) |
Theorem | dfnfc2 4454* | An alternative statement of the effective freeness of a class , when it is a set. (Contributed by Mario Carneiro, 14-Oct-2016.) (Proof shortened by JJ, 26-Jul-2021.) |
Theorem | dfnfc2OLD 4455* | Obsolete proof of dfnfc2 4454 as of 26-Jul-2021. (Contributed by Mario Carneiro, 14-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | uniun 4456 | The class union of the union of two classes. Theorem 8.3 of [Quine] p. 53. (Contributed by NM, 20-Aug-1993.) |
Theorem | uniin 4457 | The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. See uniinqs 7827 for a condition where equality holds. (Contributed by NM, 4-Dec-2003.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Theorem | uniss 4458 | Subclass relationship for class union. Theorem 61 of [Suppes] p. 39. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Theorem | ssuni 4459 | Subclass relationship for class union. (Contributed by NM, 24-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof shortened by JJ, 26-Jul-2021.) |
Theorem | ssuniOLD 4460 | Obsolete proof of ssuni 4459 as of 26-Jul-2021. (Contributed by NM, 24-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | unissi 4461 | Subclass relationship for subclass union. Inference form of uniss 4458. (Contributed by David Moews, 1-May-2017.) |
Theorem | unissd 4462 | Subclass relationship for subclass union. Deduction form of uniss 4458. (Contributed by David Moews, 1-May-2017.) |
Theorem | uni0b 4463 | The union of a set is empty iff the set is included in the singleton of the empty set. (Contributed by NM, 12-Sep-2004.) |
Theorem | uni0c 4464* | The union of a set is empty iff all of its members are empty. (Contributed by NM, 16-Aug-2006.) |
Theorem | uni0 4465 | The union of the empty set is the empty set. Theorem 8.7 of [Quine] p. 54. (Reproved without relying on ax-nul 4789 by Eric Schmidt.) (Contributed by NM, 16-Sep-1993.) (Revised by Eric Schmidt, 4-Apr-2007.) |
Theorem | csbuni 4466 | Distribute proper substitution through the union of a class. (Contributed by Alan Sare, 10-Nov-2012.) (Revised by NM, 22-Aug-2018.) |
Theorem | elssuni 4467 | An element of a class is a subclass of its union. Theorem 8.6 of [Quine] p. 54. Also the basis for Proposition 7.20 of [TakeutiZaring] p. 40. (Contributed by NM, 6-Jun-1994.) |
Theorem | unissel 4468 | Condition turning a subclass relationship for union into an equality. (Contributed by NM, 18-Jul-2006.) |
Theorem | unissb 4469* | Relationship involving membership, subset, and union. Exercise 5 of [Enderton] p. 26 and its converse. (Contributed by NM, 20-Sep-2003.) |
Theorem | uniss2 4470* | A subclass condition on the members of two classes that implies a subclass relation on their unions. Proposition 8.6 of [TakeutiZaring] p. 59. See iunss2 4565 for a generalization to indexed unions. (Contributed by NM, 22-Mar-2004.) |
Theorem | unidif 4471* | If the difference contains the largest members of , then the union of the difference is the union of . (Contributed by NM, 22-Mar-2004.) |
Theorem | ssunieq 4472* | Relationship implying union. (Contributed by NM, 10-Nov-1999.) |
Theorem | unimax 4473* | Any member of a class is the largest of those members that it includes. (Contributed by NM, 13-Aug-2002.) |
Theorem | pwuni 4474 | A class is a subclass of the power class of its union. Exercise 6(b) of [Enderton] p. 38. (Contributed by NM, 14-Oct-1996.) |
Syntax | cint 4475 | Extend class notation to include the intersection of a class (read: 'intersect '). |
Definition | df-int 4476* | Define the intersection of a class. Definition 7.35 of [TakeutiZaring] p. 44. For example, . Compare this with the intersection of two classes, df-in 3581. (Contributed by NM, 18-Aug-1993.) |
Theorem | dfint2 4477* | Alternate definition of class intersection. (Contributed by NM, 28-Jun-1998.) |
Theorem | inteq 4478 | Equality law for intersection. (Contributed by NM, 13-Sep-1999.) |
Theorem | inteqi 4479 | Equality inference for class intersection. (Contributed by NM, 2-Sep-2003.) |
Theorem | inteqd 4480 | Equality deduction for class intersection. (Contributed by NM, 2-Sep-2003.) |
Theorem | elint 4481* | Membership in class intersection. (Contributed by NM, 21-May-1994.) |
Theorem | elint2 4482* | Membership in class intersection. (Contributed by NM, 14-Oct-1999.) |
Theorem | elintg 4483* | Membership in class intersection, with the sethood requirement expressed as an antecedent. (Contributed by NM, 20-Nov-2003.) (Proof shortened by JJ, 26-Jul-2021.) |
Theorem | elintgOLD 4484* | Obsolete proof of elintg 4483 as of 26-Jul-2021. (Contributed by NM, 20-Nov-2003.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | elinti 4485 | Membership in class intersection. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Theorem | nfint 4486 | Bound-variable hypothesis builder for intersection. (Contributed by NM, 2-Feb-1997.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
Theorem | elintab 4487* | Membership in the intersection of a class abstraction. (Contributed by NM, 30-Aug-1993.) |
Theorem | elintrab 4488* | Membership in the intersection of a class abstraction. (Contributed by NM, 17-Oct-1999.) |
Theorem | elintrabg 4489* | Membership in the intersection of a class abstraction. (Contributed by NM, 17-Feb-2007.) |
Theorem | int0 4490 | The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44. (Contributed by NM, 18-Aug-1993.) (Proof shortened by JJ, 26-Jul-2021.) |
Theorem | int0OLD 4491 | Obsolete proof of int0 4490 as of 26-Jul-2021. (Contributed by NM, 18-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | intss1 4492 | An element of a class includes the intersection of the class. Exercise 4 of [TakeutiZaring] p. 44 (with correction), generalized to classes. (Contributed by NM, 18-Nov-1995.) |
Theorem | ssint 4493* | Subclass of a class intersection. Theorem 5.11(viii) of [Monk1] p. 52 and its converse. (Contributed by NM, 14-Oct-1999.) |
Theorem | ssintab 4494* | Subclass of the intersection of a class abstraction. (Contributed by NM, 31-Jul-2006.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Theorem | ssintub 4495* | Subclass of the least upper bound. (Contributed by NM, 8-Aug-2000.) |
Theorem | ssmin 4496* | Subclass of the minimum value of class of supersets. (Contributed by NM, 10-Aug-2006.) |
Theorem | intmin 4497* | Any member of a class is the smallest of those members that include it. (Contributed by NM, 13-Aug-2002.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Theorem | intss 4498 | Intersection of subclasses. (Contributed by NM, 14-Oct-1999.) (Proof shortened by OpenAI, 25-Mar-2020.) |
Theorem | intssuni 4499 | The intersection of a nonempty set is a subclass of its union. (Contributed by NM, 29-Jul-2006.) |
Theorem | ssintrab 4500* | Subclass of the intersection of a restricted class builder. (Contributed by NM, 30-Jan-2015.) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |