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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | issoi 5501* | An irreflexive, transitive, linear relation is a strict ordering. (Contributed by NM, 21-Jan-1996.) (Revised by Mario Carneiro, 9-Jul-2014.) |
⊢ (𝑥 ∈ 𝐴 → ¬ 𝑥𝑅𝑥) & ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) & ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)) ⇒ ⊢ 𝑅 Or 𝐴 | ||
Theorem | isso2i 5502* | Deduce strict ordering from its properties. (Contributed by NM, 29-Jan-1996.) (Revised by Mario Carneiro, 9-Jul-2014.) |
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ↔ ¬ (𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))) & ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ⇒ ⊢ 𝑅 Or 𝐴 | ||
Theorem | so0 5503 | 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 ∅ | ||
Theorem | somo 5504* | 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 → ∃*𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥) | ||
Syntax | wfr 5505 | Extend wff notation to include the well-founded predicate. Read: "𝑅 is a well-founded relation on 𝐴". |
wff 𝑅 Fr 𝐴 | ||
Syntax | wse 5506 | Extend wff notation to include the set-like predicate. Read: "𝑅 is set-like on 𝐴". |
wff 𝑅 Se 𝐴 | ||
Syntax | wwe 5507 | Extend wff notation to include the well-ordering predicate. Read: "𝑅 well-orders 𝐴". |
wff 𝑅 We 𝐴 | ||
Definition | df-fr 5508* | Define the well-founded relation predicate. Definition 6.24(1) of [TakeutiZaring] p. 30. For alternate definitions, see dffr2 5514 and dffr3 5956. A class is called well-founded when the membership relation E (see df-eprel 5459) is well-founded on it, that is, 𝐴 is well-founded if E Fr 𝐴 (some sources request that the membership relation be well-founded on its transitive closure). (Contributed by NM, 3-Apr-1994.) |
⊢ (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦)) | ||
Definition | df-se 5509* | Define the set-like predicate. (Contributed by Mario Carneiro, 19-Nov-2014.) |
⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V) | ||
Definition | df-we 5510 | Define the well-ordering predicate. For an alternate definition, see dfwe2 7484. (Contributed by NM, 3-Apr-1994.) |
⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴)) | ||
Theorem | fri 5511* | Property of well-founded relation (one direction of definition). (Contributed by NM, 18-Mar-1997.) |
⊢ (((𝐵 ∈ 𝐶 ∧ 𝑅 Fr 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) | ||
Theorem | seex 5512* | 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 𝐴 ∧ 𝐵 ∈ 𝐴) → {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝐵} ∈ V) | ||
Theorem | exse 5513 | Any relation on a set is set-like on it. (Contributed by Mario Carneiro, 22-Jun-2015.) |
⊢ (𝐴 ∈ 𝑉 → 𝑅 Se 𝐴) | ||
Theorem | dffr2 5514* | 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((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = ∅)) | ||
Theorem | frc 5515* | 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 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} = ∅) | ||
Theorem | frss 5516 | 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.) |
⊢ (𝐴 ⊆ 𝐵 → (𝑅 Fr 𝐵 → 𝑅 Fr 𝐴)) | ||
Theorem | sess1 5517 | Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.) |
⊢ (𝑅 ⊆ 𝑆 → (𝑆 Se 𝐴 → 𝑅 Se 𝐴)) | ||
Theorem | sess2 5518 | Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.) |
⊢ (𝐴 ⊆ 𝐵 → (𝑅 Se 𝐵 → 𝑅 Se 𝐴)) | ||
Theorem | freq1 5519 | Equality theorem for the well-founded predicate. (Contributed by NM, 9-Mar-1997.) |
⊢ (𝑅 = 𝑆 → (𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐴)) | ||
Theorem | freq2 5520 | Equality theorem for the well-founded predicate. (Contributed by NM, 3-Apr-1994.) |
⊢ (𝐴 = 𝐵 → (𝑅 Fr 𝐴 ↔ 𝑅 Fr 𝐵)) | ||
Theorem | seeq1 5521 | Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.) |
⊢ (𝑅 = 𝑆 → (𝑅 Se 𝐴 ↔ 𝑆 Se 𝐴)) | ||
Theorem | seeq2 5522 | Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.) |
⊢ (𝐴 = 𝐵 → (𝑅 Se 𝐴 ↔ 𝑅 Se 𝐵)) | ||
Theorem | nffr 5523 | Bound-variable hypothesis builder for well-founded relations. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.) |
⊢ Ⅎ𝑥𝑅 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥 𝑅 Fr 𝐴 | ||
Theorem | nfse 5524 | Bound-variable hypothesis builder for set-like relations. (Contributed by Mario Carneiro, 24-Jun-2015.) (Revised by Mario Carneiro, 14-Oct-2016.) |
⊢ Ⅎ𝑥𝑅 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥 𝑅 Se 𝐴 | ||
Theorem | nfwe 5525 | Bound-variable hypothesis builder for well-orderings. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.) |
⊢ Ⅎ𝑥𝑅 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥 𝑅 We 𝐴 | ||
Theorem | frirr 5526 | 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 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵) | ||
Theorem | fr2nr 5527 | 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 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵)) | ||
Theorem | fr0 5528 | Any relation is well-founded on the empty set. (Contributed by NM, 17-Sep-1993.) |
⊢ 𝑅 Fr ∅ | ||
Theorem | frminex 5529* | 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 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝑅 Fr 𝐴 → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑥)))) | ||
Theorem | efrirr 5530 | A well-founded class does not belong to itself. (Contributed by NM, 18-Apr-1994.) (Revised by Mario Carneiro, 22-Jun-2015.) |
⊢ ( E Fr 𝐴 → ¬ 𝐴 ∈ 𝐴) | ||
Theorem | efrn2lp 5531 | A well-founded class contains no 2-cycle loops. (Contributed by NM, 19-Apr-1994.) |
⊢ (( E Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐵)) | ||
Theorem | epse 5532 | The membership relation is set-like on any class. (This is the origin of the term "set-like": a set-like relation "acts like" the membership relation of sets and their elements.) (Contributed by Mario Carneiro, 22-Jun-2015.) |
⊢ E Se 𝐴 | ||
Theorem | tz7.2 5533 | Similar to Theorem 7.2 of [TakeutiZaring] p. 35, except that the Axiom of Regularity is not required due to the antecedent E Fr 𝐴. (Contributed by NM, 4-May-1994.) |
⊢ ((Tr 𝐴 ∧ E Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴)) | ||
Theorem | dfepfr 5534* | An alternate way of saying that the membership relation is well-founded. (Contributed by NM, 17-Feb-2004.) (Revised by Mario Carneiro, 23-Jun-2015.) |
⊢ ( E Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ 𝑦) = ∅)) | ||
Theorem | epfrc 5535* | A subset of a well-founded class has a minimal element. (Contributed by NM, 17-Feb-2004.) (Revised by David Abernethy, 22-Feb-2011.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (( E Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 (𝐵 ∩ 𝑥) = ∅) | ||
Theorem | wess 5536 | Subset theorem for the well-ordering predicate. Exercise 4 of [TakeutiZaring] p. 31. (Contributed by NM, 19-Apr-1994.) |
⊢ (𝐴 ⊆ 𝐵 → (𝑅 We 𝐵 → 𝑅 We 𝐴)) | ||
Theorem | weeq1 5537 | Equality theorem for the well-ordering predicate. (Contributed by NM, 9-Mar-1997.) |
⊢ (𝑅 = 𝑆 → (𝑅 We 𝐴 ↔ 𝑆 We 𝐴)) | ||
Theorem | weeq2 5538 | Equality theorem for the well-ordering predicate. (Contributed by NM, 3-Apr-1994.) |
⊢ (𝐴 = 𝐵 → (𝑅 We 𝐴 ↔ 𝑅 We 𝐵)) | ||
Theorem | wefr 5539 | A well-ordering is well-founded. (Contributed by NM, 22-Apr-1994.) |
⊢ (𝑅 We 𝐴 → 𝑅 Fr 𝐴) | ||
Theorem | weso 5540 | A well-ordering is a strict ordering. (Contributed by NM, 16-Mar-1997.) |
⊢ (𝑅 We 𝐴 → 𝑅 Or 𝐴) | ||
Theorem | wecmpep 5541 | The elements of a class well-ordered by membership are comparable. (Contributed by NM, 17-May-1994.) |
⊢ (( E We 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)) | ||
Theorem | wetrep 5542 | On a class well-ordered by membership, the membership predicate is transitive. (Contributed by NM, 22-Apr-1994.) |
⊢ (( E We 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧)) | ||
Theorem | wefrc 5543* | A nonempty subclass of a class well-ordered by membership has a minimal element. Special case of Proposition 6.26 of [TakeutiZaring] p. 31. (Contributed by NM, 17-Feb-2004.) |
⊢ (( E We 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 (𝐵 ∩ 𝑥) = ∅) | ||
Theorem | we0 5544 | Any relation is a well-ordering of the empty set. (Contributed by NM, 16-Mar-1997.) |
⊢ 𝑅 We ∅ | ||
Theorem | wereu 5545* | 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 ∧ (𝐵 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃!𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) | ||
Theorem | wereu2 5546* | All nonempty subclasses of a class having a well-ordered and set-like relation have a minimal element for that relation. Proposition 6.26 of [TakeutiZaring] p. 31. (Contributed by Scott Fenton, 29-Jan-2011.) (Revised by Mario Carneiro, 24-Jun-2015.) |
⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃!𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) | ||
Syntax | cxp 5547 | Extend the definition of a class to include the Cartesian product. |
class (𝐴 × 𝐵) | ||
Syntax | ccnv 5548 | Extend the definition of a class to include the converse of a class. |
class ◡𝐴 | ||
Syntax | cdm 5549 | Extend the definition of a class to include the domain of a class. |
class dom 𝐴 | ||
Syntax | crn 5550 | Extend the definition of a class to include the range of a class. |
class ran 𝐴 | ||
Syntax | cres 5551 | Extend the definition of a class to include the restriction of a class. Read: "the restriction of 𝐴 to 𝐵". |
class (𝐴 ↾ 𝐵) | ||
Syntax | cima 5552 | Extend the definition of a class to include the image of a class. Read: "the image of 𝐵 under 𝐴". |
class (𝐴 “ 𝐵) | ||
Syntax | ccom 5553 | Extend the definition of a class to include the composition of two classes. (Read: The composition of 𝐴 and 𝐵.) |
class (𝐴 ∘ 𝐵) | ||
Syntax | wrel 5554 | Extend the definition of a wff to include the relation predicate. Read: "𝐴 is a relation". |
wff Rel 𝐴 | ||
Definition | df-xp 5555* | 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 28143). Another example is that the set of rational numbers are defined in df-q 12338 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〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)} | ||
Definition | df-rel 5556 | Define the relation predicate. Definition 6.4(1) of [TakeutiZaring] p. 23. For alternate definitions, see dfrel2 6040 and dfrel3 6049. (Contributed by NM, 1-Aug-1994.) |
⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V)) | ||
Definition | df-cnv 5557* | 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 (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴), as proven in brcnv 5747 (see df-br 5059 and df-rel 5556 for more on relations). For example, ◡{〈2, 6〉, 〈3, 9〉} = {〈6, 2〉, 〈9, 3〉} (ex-cnv 28144). 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 "minus one". The term "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} | ||
Definition | df-co 5558* | Define the composition of two classes. Definition 6.6(3) of [TakeutiZaring] p. 24. For example, ((exp ∘ cos)‘0) = e (ex-co 28145) because (cos‘0) = 1 (see cos0 15493) and (exp‘1) = e (see df-e 15412). 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.) |
⊢ (𝐴 ∘ 𝐵) = {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)} | ||
Definition | df-dm 5559* | 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 28146). Another example is the domain of the complex arctangent, (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ -i ∧ 𝐴 ≠ i)) (for proof see atandm 25381). Contrast with range (defined in df-rn 5560). For alternate definitions see dfdm2 6126, dfdm3 5752, and dfdm4 5758. The notation "dom " is used by Enderton; other authors sometimes use script D. (Contributed by NM, 1-Aug-1994.) |
⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} | ||
Definition | df-rn 5560 | Define the range of a class. For example, 𝐹 = {〈2, 6〉, 〈3, 9〉} → ran 𝐹 = {6, 9} (ex-rn 28147). Contrast with domain (defined in df-dm 5559). For alternate definitions, see dfrn2 5753, dfrn3 5754, and dfrn4 6053. The notation "ran " is used by Enderton. The range of a function is often also called "the image of the function" (see definition in [Lang] p. ix), which can be justified by imadmrn 5933. Not to be confused with "codomain" (see df-f 6353), which may be a superset/superclass of the range (see frn 6514). (Contributed by NM, 1-Aug-1994.) |
⊢ ran 𝐴 = dom ◡𝐴 | ||
Definition | df-res 5561 | Define the restriction of a class. Definition 6.6(1) of [TakeutiZaring] p. 24. For example, the expression (exp ↾ ℝ) (used in reeff1 15463) means "the exponential function e to the x, but the exponent x must be in the reals" (df-ef 15411 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}) → (𝐹 ↾ 𝐵) = {〈2, 6〉} (ex-res 28148). We do not introduce a special syntax for the corestriction of a class: it will be expressed either as the intersection (𝐴 ∩ (V × 𝐵)) or as the converse of the restricted converse (see cnvrescnv 6046). (Contributed by NM, 2-Aug-1994.) |
⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V)) | ||
Definition | df-ima 5562 | 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}) → (𝐹 “ 𝐵) = {6} (ex-ima 28149). Contrast with restriction (df-res 5561) and range (df-rn 5560). For an alternate definition, see dfima2 5925. (Contributed by NM, 2-Aug-1994.) |
⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) | ||
Theorem | xpeq1 5563 | Equality theorem for Cartesian product. (Contributed by NM, 4-Jul-1994.) |
⊢ (𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶)) | ||
Theorem | xpss12 5564 | 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.) |
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (𝐴 × 𝐶) ⊆ (𝐵 × 𝐷)) | ||
Theorem | xpss 5565 | 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) | ||
Theorem | inxpssres 5566 | Intersection with a Cartesian product is a subclass of restriction. (Contributed by Peter Mazsa, 19-Jul-2019.) |
⊢ (𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝑅 ↾ 𝐴) | ||
Theorem | relxp 5567 | A Cartesian product is a relation. Theorem 3.13(i) of [Monk1] p. 37. (Contributed by NM, 2-Aug-1994.) |
⊢ Rel (𝐴 × 𝐵) | ||
Theorem | xpss1 5568 | Subset relation for Cartesian product. (Contributed by Jeff Hankins, 30-Aug-2009.) |
⊢ (𝐴 ⊆ 𝐵 → (𝐴 × 𝐶) ⊆ (𝐵 × 𝐶)) | ||
Theorem | xpss2 5569 | Subset relation for Cartesian product. (Contributed by Jeff Hankins, 30-Aug-2009.) |
⊢ (𝐴 ⊆ 𝐵 → (𝐶 × 𝐴) ⊆ (𝐶 × 𝐵)) | ||
Theorem | xpeq2 5570 | Equality theorem for Cartesian product. (Contributed by NM, 5-Jul-1994.) |
⊢ (𝐴 = 𝐵 → (𝐶 × 𝐴) = (𝐶 × 𝐵)) | ||
Theorem | elxpi 5571* | Membership in a Cartesian product. Uses fewer axioms than elxp 5572. (Contributed by NM, 4-Jul-1994.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) | ||
Theorem | elxp 5572* | Membership in a Cartesian product. (Contributed by NM, 4-Jul-1994.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) | ||
Theorem | elxp2 5573* | Membership in a Cartesian product. (Contributed by NM, 23-Feb-2004.) (Proof shortened by JJ, 13-Aug-2021.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐴 = 〈𝑥, 𝑦〉) | ||
Theorem | xpeq12 5574 | Equality theorem for Cartesian product. (Contributed by FL, 31-Aug-2009.) |
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷)) | ||
Theorem | xpeq1i 5575 | Equality inference for Cartesian product. (Contributed by NM, 21-Dec-2008.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 × 𝐶) = (𝐵 × 𝐶) | ||
Theorem | xpeq2i 5576 | Equality inference for Cartesian product. (Contributed by NM, 21-Dec-2008.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 × 𝐴) = (𝐶 × 𝐵) | ||
Theorem | xpeq12i 5577 | Equality inference for Cartesian product. (Contributed by FL, 31-Aug-2009.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ (𝐴 × 𝐶) = (𝐵 × 𝐷) | ||
Theorem | xpeq1d 5578 | Equality deduction for Cartesian product. (Contributed by Jeff Madsen, 17-Jun-2010.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 × 𝐶) = (𝐵 × 𝐶)) | ||
Theorem | xpeq2d 5579 | Equality deduction for Cartesian product. (Contributed by Jeff Madsen, 17-Jun-2010.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 × 𝐴) = (𝐶 × 𝐵)) | ||
Theorem | xpeq12d 5580 | Equality deduction for Cartesian product. (Contributed by NM, 8-Dec-2013.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴 × 𝐶) = (𝐵 × 𝐷)) | ||
Theorem | sqxpeqd 5581 | 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.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 × 𝐴) = (𝐵 × 𝐵)) | ||
Theorem | nfxp 5582 | Bound-variable hypothesis builder for Cartesian product. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 15-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥(𝐴 × 𝐵) | ||
Theorem | 0nelxp 5583 | 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) | ||
Theorem | 0nelelxp 5584 | A member of a Cartesian product (ordered pair) doesn't contain the empty set. (Contributed by NM, 15-Dec-2008.) |
⊢ (𝐶 ∈ (𝐴 × 𝐵) → ¬ ∅ ∈ 𝐶) | ||
Theorem | opelxp 5585 | 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) ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) | ||
Theorem | opelxpi 5586 | Ordered pair membership in a Cartesian product (implication). (Contributed by NM, 28-May-1995.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷)) | ||
Theorem | opelxpd 5587 | Ordered pair membership in a Cartesian product, deduction form. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
⊢ (𝜑 → 𝐴 ∈ 𝐶) & ⊢ (𝜑 → 𝐵 ∈ 𝐷) ⇒ ⊢ (𝜑 → 〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷)) | ||
Theorem | opelvv 5588 | Ordered pair membership in the universal class of ordered pairs. (Contributed by NM, 22-Aug-2013.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ 〈𝐴, 𝐵〉 ∈ (V × V) | ||
Theorem | opelvvg 5589 | Ordered pair membership in the universal class of ordered pairs. (Contributed by Mario Carneiro, 3-May-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 〈𝐴, 𝐵〉 ∈ (V × V)) | ||
Theorem | opelxp1 5590 | 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) → 𝐴 ∈ 𝐶) | ||
Theorem | opelxp2 5591 | 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) → 𝐵 ∈ 𝐷) | ||
Theorem | otelxp1 5592 | 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) → 𝐴 ∈ 𝑅) | ||
Theorem | otel3xp 5593 | An ordered triple is an element of a doubled Cartesian product. (Contributed by Alexander van der Vekens, 26-Feb-2018.) |
⊢ ((𝑇 = 〈𝐴, 𝐵, 𝐶〉 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍)) → 𝑇 ∈ ((𝑋 × 𝑌) × 𝑍)) | ||
Theorem | rabxp 5594* | Membership in a class builder restricted to a Cartesian product. (Contributed by NM, 20-Feb-2014.) |
⊢ (𝑥 = 〈𝑦, 𝑧〉 → (𝜑 ↔ 𝜓)) ⇒ ⊢ {𝑥 ∈ (𝐴 × 𝐵) ∣ 𝜑} = {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ∧ 𝜓)} | ||
Theorem | brxp 5595 | Binary relation on a Cartesian product. (Contributed by NM, 22-Apr-2004.) |
⊢ (𝐴(𝐶 × 𝐷)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) | ||
Theorem | pwvrel 5596 | A set is a binary relation if and only if it belongs to the powerclass of the cartesian square of the universal class. (Contributed by Peter Mazsa, 14-Jun-2018.) (Revised by BJ, 16-Dec-2023.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 (V × V) ↔ Rel 𝐴)) | ||
Theorem | pwvabrel 5597 | The powerclass of the cartesian square of the universal class is the class of all sets which are binary relations. (Contributed by BJ, 21-Dec-2023.) |
⊢ 𝒫 (V × V) = {𝑥 ∣ Rel 𝑥} | ||
Theorem | brrelex12 5598 | Two classes related by a binary relation are sets. (Contributed by Mario Carneiro, 26-Apr-2015.) |
⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | ||
Theorem | brrelex1 5599 | If two classes are related by a binary relation, then the first class is a set. (Contributed by NM, 18-May-2004.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V) | ||
Theorem | brrelex2 5600 | If two classes are related by a binary relation, then the second class is a set. (Contributed by Mario Carneiro, 26-Apr-2015.) |
⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V) |
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