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Theorem List for Metamath Proof Explorer - 5201-5300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremissoi 5201* An irreflexive, transitive, linear relation is a strict ordering. (Contributed by NM, 21-Jan-1996.) (Revised by Mario Carneiro, 9-Jul-2014.)
(𝑥𝐴 → ¬ 𝑥𝑅𝑥)    &   ((𝑥𝐴𝑦𝐴𝑧𝐴) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))    &   ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))       𝑅 Or 𝐴
 
Theoremisso2i 5202* Deduce strict ordering from its properties. (Contributed by NM, 29-Jan-1996.) (Revised by Mario Carneiro, 9-Jul-2014.)
((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦 ↔ ¬ (𝑥 = 𝑦𝑦𝑅𝑥)))    &   ((𝑥𝐴𝑦𝐴𝑧𝐴) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))       𝑅 Or 𝐴
 
Theoremso0 5203 Any relation is a strict ordering of the empty set. (Contributed by NM, 16-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
𝑅 Or ∅
 
Theoremsomo 5204* A totally ordered set has at most one minimal element. (Contributed by Mario Carneiro, 24-Jun-2015.) (Revised by NM, 16-Jun-2017.)
(𝑅 Or 𝐴 → ∃*𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥)
 
2.3.9  Founded and well-ordering relations
 
Syntaxwfr 5205 Extend wff notation to include the well-founded predicate. Read: ' 𝑅 is a well-founded relation on 𝐴.'
wff 𝑅 Fr 𝐴
 
Syntaxwse 5206 Extend wff notation to include the set-like predicate. Read: ' 𝑅 is set-like on 𝐴.'
wff 𝑅 Se 𝐴
 
Syntaxwwe 5207 Extend wff notation to include the well-ordering predicate. Read: ' 𝑅 well-orders 𝐴.'
wff 𝑅 We 𝐴
 
Definitiondf-fr 5208* Define the well-founded relation predicate. Definition 6.24(1) of [TakeutiZaring] p. 30. For alternate definitions, see dffr2 5214 and dffr3 5639. (Contributed by NM, 3-Apr-1994.)
(𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
 
Definitiondf-se 5209* Define the set-like predicate. (Contributed by Mario Carneiro, 19-Nov-2014.)
(𝑅 Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V)
 
Definitiondf-we 5210 Define the well-ordering predicate. For an alternate definition, see dfwe2 7128. (Contributed by NM, 3-Apr-1994.)
(𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴𝑅 Or 𝐴))
 
Theoremfri 5211* Property of well-founded relation (one direction of definition). (Contributed by NM, 18-Mar-1997.)
(((𝐵𝐶𝑅 Fr 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
 
Theoremseex 5212* The 𝑅-preimage of an element of the base set in a set-like relation is a set. (Contributed by Mario Carneiro, 19-Nov-2014.)
((𝑅 Se 𝐴𝐵𝐴) → {𝑥𝐴𝑥𝑅𝐵} ∈ V)
 
Theoremexse 5213 Any relation on a set is set-like on it. (Contributed by Mario Carneiro, 22-Jun-2015.)
(𝐴𝑉𝑅 Se 𝐴)
 
Theoremdffr2 5214* Alternate definition of well-founded relation. Similar to Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 17-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof shortened by Mario Carneiro, 23-Jun-2015.)
(𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅))
 
Theoremfrc 5215* Property of well-founded relation (one direction of definition using class variables). (Contributed by NM, 17-Feb-2004.) (Revised by Mario Carneiro, 19-Nov-2014.)
𝐵 ∈ V       ((𝑅 Fr 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵 {𝑦𝐵𝑦𝑅𝑥} = ∅)
 
Theoremfrss 5216 Subset theorem for the well-founded predicate. Exercise 1 of [TakeutiZaring] p. 31. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(𝐴𝐵 → (𝑅 Fr 𝐵𝑅 Fr 𝐴))
 
Theoremsess1 5217 Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
(𝑅𝑆 → (𝑆 Se 𝐴𝑅 Se 𝐴))
 
Theoremsess2 5218 Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
(𝐴𝐵 → (𝑅 Se 𝐵𝑅 Se 𝐴))
 
Theoremfreq1 5219 Equality theorem for the well-founded predicate. (Contributed by NM, 9-Mar-1997.)
(𝑅 = 𝑆 → (𝑅 Fr 𝐴𝑆 Fr 𝐴))
 
Theoremfreq2 5220 Equality theorem for the well-founded predicate. (Contributed by NM, 3-Apr-1994.)
(𝐴 = 𝐵 → (𝑅 Fr 𝐴𝑅 Fr 𝐵))
 
Theoremseeq1 5221 Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
(𝑅 = 𝑆 → (𝑅 Se 𝐴𝑆 Se 𝐴))
 
Theoremseeq2 5222 Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
(𝐴 = 𝐵 → (𝑅 Se 𝐴𝑅 Se 𝐵))
 
Theoremnffr 5223 Bound-variable hypothesis builder for well-founded relations. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)
𝑥𝑅    &   𝑥𝐴       𝑥 𝑅 Fr 𝐴
 
Theoremnfse 5224 Bound-variable hypothesis builder for set-like relations. (Contributed by Mario Carneiro, 24-Jun-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)
𝑥𝑅    &   𝑥𝐴       𝑥 𝑅 Se 𝐴
 
Theoremnfwe 5225 Bound-variable hypothesis builder for well-orderings. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)
𝑥𝑅    &   𝑥𝐴       𝑥 𝑅 We 𝐴
 
Theoremfrirr 5226 A well-founded relation is irreflexive. Special case of Proposition 6.23 of [TakeutiZaring] p. 30. (Contributed by NM, 2-Jan-1994.) (Revised by Mario Carneiro, 22-Jun-2015.)
((𝑅 Fr 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)
 
Theoremfr2nr 5227 A well-founded relation has no 2-cycle loops. Special case of Proposition 6.23 of [TakeutiZaring] p. 30. (Contributed by NM, 30-May-1994.) (Revised by Mario Carneiro, 22-Jun-2015.)
((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))
 
Theoremfr0 5228 Any relation is well-founded on the empty set. (Contributed by NM, 17-Sep-1993.)
𝑅 Fr ∅
 
Theoremfrminex 5229* If an element of a well-founded set satisfies a property 𝜑, then there is a minimal element that satisfies 𝜑. (Contributed by Jeff Madsen, 18-Jun-2010.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
𝐴 ∈ V    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝑅 Fr 𝐴 → (∃𝑥𝐴 𝜑 → ∃𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 (𝜓 → ¬ 𝑦𝑅𝑥))))
 
Theoremefrirr 5230 Irreflexivity of the epsilon relation: a class founded by epsilon is not a member of itself. (Contributed by NM, 18-Apr-1994.) (Revised by Mario Carneiro, 22-Jun-2015.)
( E Fr 𝐴 → ¬ 𝐴𝐴)
 
Theoremefrn2lp 5231 A set founded by epsilon contains no 2-cycle loops. (Contributed by NM, 19-Apr-1994.)
(( E Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝐶𝐶𝐵))
 
Theoremepse 5232 The epsilon relation is set-like on any class. (This is the origin of the term "set-like": a set-like relation "acts like" the epsilon relation of sets and their elements.) (Contributed by Mario Carneiro, 22-Jun-2015.)
E Se 𝐴
 
Theoremtz7.2 5233 Similar to Theorem 7.2 of [TakeutiZaring] p. 35, of except that the Axiom of Regularity is not required due to antecedent E Fr 𝐴. (Contributed by NM, 4-May-1994.)
((Tr 𝐴 ∧ E Fr 𝐴𝐵𝐴) → (𝐵𝐴𝐵𝐴))
 
Theoremdfepfr 5234* An alternate way of saying that the epsilon relation is well-founded. (Contributed by NM, 17-Feb-2004.) (Revised by Mario Carneiro, 23-Jun-2015.)
( E Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥𝑦) = ∅))
 
Theoremepfrc 5235* A subset of an epsilon-founded class has a minimal element. (Contributed by NM, 17-Feb-2004.) (Revised by David Abernethy, 22-Feb-2011.)
𝐵 ∈ V       (( E Fr 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵 (𝐵𝑥) = ∅)
 
Theoremwess 5236 Subset theorem for the well-ordering predicate. Exercise 4 of [TakeutiZaring] p. 31. (Contributed by NM, 19-Apr-1994.)
(𝐴𝐵 → (𝑅 We 𝐵𝑅 We 𝐴))
 
Theoremweeq1 5237 Equality theorem for the well-ordering predicate. (Contributed by NM, 9-Mar-1997.)
(𝑅 = 𝑆 → (𝑅 We 𝐴𝑆 We 𝐴))
 
Theoremweeq2 5238 Equality theorem for the well-ordering predicate. (Contributed by NM, 3-Apr-1994.)
(𝐴 = 𝐵 → (𝑅 We 𝐴𝑅 We 𝐵))
 
Theoremwefr 5239 A well-ordering is well-founded. (Contributed by NM, 22-Apr-1994.)
(𝑅 We 𝐴𝑅 Fr 𝐴)
 
Theoremweso 5240 A well-ordering is a strict ordering. (Contributed by NM, 16-Mar-1997.)
(𝑅 We 𝐴𝑅 Or 𝐴)
 
Theoremwecmpep 5241 The elements of an epsilon well-ordering are comparable. (Contributed by NM, 17-May-1994.)
(( E We 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
 
Theoremwetrep 5242 An epsilon well-ordering is a transitive relation. (Contributed by NM, 22-Apr-1994.)
(( E We 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
 
Theoremwefrc 5243* A nonempty (possibly proper) subclass of a class well-ordered by E has a minimal element. Special case of Proposition 6.26 of [TakeutiZaring] p. 31. (Contributed by NM, 17-Feb-2004.)
(( E We 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵 (𝐵𝑥) = ∅)
 
Theoremwe0 5244 Any relation is a well-ordering of the empty set. (Contributed by NM, 16-Mar-1997.)
𝑅 We ∅
 
Theoremwereu 5245* A subset of a well-ordered set has a unique minimal element. (Contributed by NM, 18-Mar-1997.) (Revised by Mario Carneiro, 28-Apr-2015.)
((𝑅 We 𝐴 ∧ (𝐵𝑉𝐵𝐴𝐵 ≠ ∅)) → ∃!𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
 
Theoremwereu2 5246* All nonempty (possibly proper) subclasses of 𝐴, which has a well-founded relation 𝑅, have 𝑅-minimal elements. Proposition 6.26 of [TakeutiZaring] p. 31. (Contributed by Scott Fenton, 29-Jan-2011.) (Revised by Mario Carneiro, 24-Jun-2015.)
(((𝑅 We 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃!𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
 
2.3.10  Relations
 
Syntaxcxp 5247 Extend the definition of a class to include the Cartesian product.
class (𝐴 × 𝐵)
 
Syntaxccnv 5248 Extend the definition of a class to include the converse of a class.
class 𝐴
 
Syntaxcdm 5249 Extend the definition of a class to include the domain of a class.
class dom 𝐴
 
Syntaxcrn 5250 Extend the definition of a class to include the range of a class.
class ran 𝐴
 
Syntaxcres 5251 Extend the definition of a class to include the restriction of a class. (Read: The restriction of 𝐴 to 𝐵.)
class (𝐴𝐵)
 
Syntaxcima 5252 Extend the definition of a class to include the image of a class. (Read: The image of 𝐵 under 𝐴.)
class (𝐴𝐵)
 
Syntaxccom 5253 Extend the definition of a class to include the composition of two classes. (Read: The composition of 𝐴 and 𝐵.)
class (𝐴𝐵)
 
Syntaxwrel 5254 Extend the definition of a wff to include the relation predicate. (Read: 𝐴 is a relation.)
wff Rel 𝐴
 
Definitiondf-xp 5255* Define the Cartesian product of two classes. This is also sometimes called the "cross product" but that term also has other meanings; we intentionally choose a less ambiguous term. Definition 9.11 of [Quine] p. 64. For example, ({1, 5} × {2, 7}) = ({⟨1, 2⟩, ⟨1, 7⟩} ∪ {⟨5, 2⟩, ⟨5, 7⟩}) (ex-xp 27635). Another example is that the set of rational numbers are defined in df-q 11992 using the Cartesian product (ℤ × ℕ); the left- and right-hand sides of the Cartesian product represent the top (integer) and bottom (natural) numbers of a fraction. (Contributed by NM, 4-Jul-1994.)
(𝐴 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)}
 
Definitiondf-rel 5256 Define the relation predicate. Definition 6.4(1) of [TakeutiZaring] p. 23. For alternate definitions, see dfrel2 5724 and dfrel3 5733. (Contributed by NM, 1-Aug-1994.)
(Rel 𝐴𝐴 ⊆ (V × V))
 
Definitiondf-cnv 5257* Define the converse of a class. Definition 9.12 of [Quine] p. 64. The converse of a binary relation swaps its arguments, i.e., if 𝐴 ∈ V and 𝐵 ∈ V then (𝐴𝑅𝐵𝐵𝑅𝐴), as proven in brcnv 5443 (see df-br 4787 and df-rel 5256 for more on relations). For example, {⟨2, 6⟩, ⟨3, 9⟩} = {⟨6, 2⟩, ⟨9, 3⟩} (ex-cnv 27636). We use Quine's breve accent (smile) notation. Like Quine, we use it as a prefix, which eliminates the need for parentheses. Many authors use the postfix superscript "to the minus one." "Converse" is Quine's terminology; some authors call it "inverse," especially when the argument is a function. (Contributed by NM, 4-Jul-1994.)
𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
 
Definitiondf-co 5258* Define the composition of two classes. Definition 6.6(3) of [TakeutiZaring] p. 24. For example, ((exp ∘ cos)‘0) = e (ex-co 27637) because (cos‘0) = 1 (see cos0 15086) and (exp‘1) = e (see df-e 15005). Note that Definition 7 of [Suppes] p. 63 reverses 𝐴 and 𝐵, uses / instead of , and calls the operation "relative product." (Contributed by NM, 4-Jul-1994.)
(𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦)}
 
Definitiondf-dm 5259* Define the domain of a class. Definition 3 of [Suppes] p. 59. For example, 𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → dom 𝐹 = {2, 3} (ex-dm 27638). Another example is the domain of the complex arctangent, (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ -i ∧ 𝐴 ≠ i)) (for proof see atandm 24824). Contrast with range (defined in df-rn 5260). For alternate definitions see dfdm2 5811, dfdm3 5448, and dfdm4 5454. The notation "dom " is used by Enderton; other authors sometimes use script D. (Contributed by NM, 1-Aug-1994.)
dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
 
Definitiondf-rn 5260 Define the range of a class. For example, 𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → ran 𝐹 = {6, 9} (ex-rn 27639). Contrast with domain (defined in df-dm 5259). For alternate definitions, see dfrn2 5449, dfrn3 5450, and dfrn4 5736. The notation "ran " is used by Enderton; other authors sometimes use script R or script W. (Contributed by NM, 1-Aug-1994.)
ran 𝐴 = dom 𝐴
 
Definitiondf-res 5261 Define the restriction of a class. Definition 6.6(1) of [TakeutiZaring] p. 24. For example, the expression (exp ↾ ℝ) (used in reeff1 15056) means "the exponential function e to the x, but the exponent x must be in the reals" (df-ef 15004 defines the exponential function, which normally allows the exponent to be a complex number). Another example is that (𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} 𝐵 = {1, 2}) → (𝐹𝐵) = {⟨2, 6⟩} (ex-res 27640). (Contributed by NM, 2-Aug-1994.)
(𝐴𝐵) = (𝐴 ∩ (𝐵 × V))
 
Definitiondf-ima 5262 Define the image of a class (as restricted by another class). Definition 6.6(2) of [TakeutiZaring] p. 24. For example, (𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} ∧ 𝐵 = {1, 2}) → (𝐹𝐵) = {6} (ex-ima 27641). Contrast with restriction (df-res 5261) and range (df-rn 5260). For an alternate definition, see dfima2 5609. (Contributed by NM, 2-Aug-1994.)
(𝐴𝐵) = ran (𝐴𝐵)
 
Theoremxpeq1 5263 Equality theorem for Cartesian product. (Contributed by NM, 4-Jul-1994.)
(𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶))
 
Theoremxpss12 5264 Subset theorem for Cartesian product. Generalization of Theorem 101 of [Suppes] p. 52. (Contributed by NM, 26-Aug-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
((𝐴𝐵𝐶𝐷) → (𝐴 × 𝐶) ⊆ (𝐵 × 𝐷))
 
Theoremxpss 5265 A Cartesian product is included in the ordered pair universe. Exercise 3 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.)
(𝐴 × 𝐵) ⊆ (V × V)
 
Theoremrelxp 5266 A Cartesian product is a relation. Theorem 3.13(i) of [Monk1] p. 37. (Contributed by NM, 2-Aug-1994.)
Rel (𝐴 × 𝐵)
 
Theoremxpss1 5267 Subset relation for Cartesian product. (Contributed by Jeff Hankins, 30-Aug-2009.)
(𝐴𝐵 → (𝐴 × 𝐶) ⊆ (𝐵 × 𝐶))
 
Theoremxpss2 5268 Subset relation for Cartesian product. (Contributed by Jeff Hankins, 30-Aug-2009.)
(𝐴𝐵 → (𝐶 × 𝐴) ⊆ (𝐶 × 𝐵))
 
Theoremxpeq2 5269 Equality theorem for Cartesian product. (Contributed by NM, 5-Jul-1994.)
(𝐴 = 𝐵 → (𝐶 × 𝐴) = (𝐶 × 𝐵))
 
Theoremelxpi 5270* Membership in a Cartesian product. Uses fewer axioms than elxp 5271. (Contributed by NM, 4-Jul-1994.)
(𝐴 ∈ (𝐵 × 𝐶) → ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
 
Theoremelxp 5271* Membership in a Cartesian product. (Contributed by NM, 4-Jul-1994.)
(𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
 
Theoremelxp2 5272* Membership in a Cartesian product. (Contributed by NM, 23-Feb-2004.) (Proof shortened by JJ, 13-Aug-2021.)
(𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝐵𝑦𝐶 𝐴 = ⟨𝑥, 𝑦⟩)
 
Theoremelxp2OLD 5273* Obsolete proof of elxp2 5272 as of 13-Aug-2021. (Contributed by NM, 23-Feb-2004.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝐵𝑦𝐶 𝐴 = ⟨𝑥, 𝑦⟩)
 
Theoremxpeq12 5274 Equality theorem for Cartesian product. (Contributed by FL, 31-Aug-2009.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷))
 
Theoremxpeq1i 5275 Equality inference for Cartesian product. (Contributed by NM, 21-Dec-2008.)
𝐴 = 𝐵       (𝐴 × 𝐶) = (𝐵 × 𝐶)
 
Theoremxpeq2i 5276 Equality inference for Cartesian product. (Contributed by NM, 21-Dec-2008.)
𝐴 = 𝐵       (𝐶 × 𝐴) = (𝐶 × 𝐵)
 
Theoremxpeq12i 5277 Equality inference for Cartesian product. (Contributed by FL, 31-Aug-2009.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴 × 𝐶) = (𝐵 × 𝐷)
 
Theoremxpeq1d 5278 Equality deduction for Cartesian product. (Contributed by Jeff Madsen, 17-Jun-2010.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴 × 𝐶) = (𝐵 × 𝐶))
 
Theoremxpeq2d 5279 Equality deduction for Cartesian product. (Contributed by Jeff Madsen, 17-Jun-2010.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶 × 𝐴) = (𝐶 × 𝐵))
 
Theoremxpeq12d 5280 Equality deduction for Cartesian product. (Contributed by NM, 8-Dec-2013.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴 × 𝐶) = (𝐵 × 𝐷))
 
Theoremsqxpeqd 5281 Equality deduction for a Cartesian square, see Wikipedia "Cartesian product", https://en.wikipedia.org/wiki/Cartesian_product#n-ary_Cartesian_power. (Contributed by AV, 13-Jan-2020.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴 × 𝐴) = (𝐵 × 𝐵))
 
Theoremnfxp 5282 Bound-variable hypothesis builder for Cartesian product. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝐴    &   𝑥𝐵       𝑥(𝐴 × 𝐵)
 
Theorem0nelxp 5283 The empty set is not a member of a Cartesian product. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 26-Apr-2015.) (Proof shortened by JJ, 13-Aug-2021.)
¬ ∅ ∈ (𝐴 × 𝐵)
 
Theorem0nelxpOLD 5284 Obsolete proof of 0nelxp 5283 as of 13-Aug-2021. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 26-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
¬ ∅ ∈ (𝐴 × 𝐵)
 
Theorem0nelelxp 5285 A member of a Cartesian product (ordered pair) doesn't contain the empty set. (Contributed by NM, 15-Dec-2008.)
(𝐶 ∈ (𝐴 × 𝐵) → ¬ ∅ ∈ 𝐶)
 
Theoremopelxp 5286 Ordered pair membership in a Cartesian product. (Contributed by NM, 15-Nov-1994.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
(⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) ↔ (𝐴𝐶𝐵𝐷))
 
Theorembrxp 5287 Binary relation on a Cartesian product. (Contributed by NM, 22-Apr-2004.)
(𝐴(𝐶 × 𝐷)𝐵 ↔ (𝐴𝐶𝐵𝐷))
 
Theoremopelxpi 5288 Ordered pair membership in a Cartesian product (implication). (Contributed by NM, 28-May-1995.)
((𝐴𝐶𝐵𝐷) → ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷))
 
Theoremopelxpd 5289 Ordered pair membership in a Cartesian product, deduction form. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝐶)    &   (𝜑𝐵𝐷)       (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷))
 
Theoremopelxp1 5290 The first member of an ordered pair of classes in a Cartesian product belongs to first Cartesian product argument. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
(⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) → 𝐴𝐶)
 
Theoremopelxp2 5291 The second member of an ordered pair of classes in a Cartesian product belongs to second Cartesian product argument. (Contributed by Mario Carneiro, 26-Apr-2015.)
(⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) → 𝐵𝐷)
 
Theoremotelxp1 5292 The first member of an ordered triple of classes in a Cartesian product belongs to first Cartesian product argument. (Contributed by NM, 28-May-2008.)
(⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ ((𝑅 × 𝑆) × 𝑇) → 𝐴𝑅)
 
Theoremotel3xp 5293 An ordered triple is an element of a doubled Cartesian product. (Contributed by Alexander van der Vekens, 26-Feb-2018.)
((𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ ∧ (𝐴𝑋𝐵𝑌𝐶𝑍)) → 𝑇 ∈ ((𝑋 × 𝑌) × 𝑍))
 
Theoremrabxp 5294* Membership in a class builder restricted to a Cartesian product. (Contributed by NM, 20-Feb-2014.)
(𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))       {𝑥 ∈ (𝐴 × 𝐵) ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧𝐵𝜓)}
 
Theorembrrelex12 5295 A true binary relation on a relation implies the arguments are sets. (This is a property of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.)
((Rel 𝑅𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
 
Theorembrrelex 5296 A true binary relation on a relation implies the first argument is a set. (This is a property of our ordered pair definition.) (Contributed by NM, 18-May-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ V)
 
Theorembrrelex2 5297 A true binary relation on a relation implies the second argument is a set. (This is a property of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.)
((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ V)
 
Theorembrrelexi 5298 The first argument of a binary relation exists. (An artifact of our ordered pair definition.) (Contributed by NM, 4-Jun-1998.)
Rel 𝑅       (𝐴𝑅𝐵𝐴 ∈ V)
 
Theorembrrelex2i 5299 The second argument of a binary relation exists. (An artifact of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.)
Rel 𝑅       (𝐴𝑅𝐵𝐵 ∈ V)
 
Theoremnprrel12 5300 Proper classes are not related via any relation. (Contributed by AV, 29-Oct-2021.)
Rel 𝑅       (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ 𝐴𝑅𝐵)
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