P. 48 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 4701-4800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	finds2 4701*	Principle of Finite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last two are the basis and the induction step. Theorem Schema 22 of [Suppes] p. 136. (Contributed by NM, 29-Nov-2002.)
⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))    &   ⊢ (𝜏 → 𝜓)    &   ⊢ (𝑦 ∈ ω → (𝜏 → (𝜒 → 𝜃)))    ⇒   ⊢ (𝑥 ∈ ω → (𝜏 → 𝜑))
Theorem	finds1 4702*	Principle of Finite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last two are the basis and the induction step. Theorem Schema 22 of [Suppes] p. 136. (Contributed by NM, 22-Mar-2006.)
⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))    &   ⊢ 𝜓    &   ⊢ (𝑦 ∈ ω → (𝜒 → 𝜃))    ⇒   ⊢ (𝑥 ∈ ω → 𝜑)
Theorem	findes 4703	Finite induction with explicit substitution. The first hypothesis is the basis and the second is the induction step. Theorem Schema 22 of [Suppes] p. 136. This is an alternative for Metamath 100 proof #74. (Contributed by Raph Levien, 9-Jul-2003.)
⊢ [∅ / 𝑥]𝜑    &   ⊢ (𝑥 ∈ ω → (𝜑 → [suc 𝑥 / 𝑥]𝜑))    ⇒   ⊢ (𝑥 ∈ ω → 𝜑)
2.6.5  The Natural Numbers (continued)
Theorem	nn0suc 4704*	A natural number is either 0 or a successor. Similar theorems for arbitrary sets or real numbers will not be provable (without the law of the excluded middle), but equality of natural numbers is decidable. (Contributed by NM, 27-May-1998.)
⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
Theorem	elomssom 4705	A natural number ordinal is, as a set, included in the set of natural number ordinals. (Contributed by NM, 21-Jun-1998.) Extract this result from the previous proof of elnn 4706. (Revised by BJ, 7-Aug-2024.)
⊢ (𝐴 ∈ ω → 𝐴 ⊆ ω)
Theorem	elnn 4706	A member of a natural number is a natural number. (Contributed by NM, 21-Jun-1998.)
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝐴 ∈ ω)
Theorem	ordom 4707	Omega is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. (Contributed by NM, 18-Oct-1995.)
⊢ Ord ω
Theorem	omelon2 4708	Omega is an ordinal number. (Contributed by Mario Carneiro, 30-Jan-2013.)
⊢ (ω ∈ V → ω ∈ On)
Theorem	omelon 4709	Omega is an ordinal number. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.)
⊢ ω ∈ On
Theorem	nnon 4710	A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)
⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
Theorem	nnoni 4711	A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)
⊢ 𝐴 ∈ ω    ⇒   ⊢ 𝐴 ∈ On
Theorem	nnord 4712	A natural number is ordinal. (Contributed by NM, 17-Oct-1995.)
⊢ (𝐴 ∈ ω → Ord 𝐴)
Theorem	omsson 4713	Omega is a subset of On. (Contributed by NM, 13-Jun-1994.)
⊢ ω ⊆ On
Theorem	limom 4714	Omega is a limit ordinal. Theorem 2.8 of [BellMachover] p. 473. (Contributed by NM, 26-Mar-1995.) (Proof rewritten by Jim Kingdon, 5-Jan-2019.)
⊢ Lim ω
Theorem	peano2b 4715	A class belongs to omega iff its successor does. (Contributed by NM, 3-Dec-1995.)
⊢ (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)
Theorem	nnsuc 4716*	A nonzero natural number is a successor. (Contributed by NM, 18-Feb-2004.)
⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥)
Theorem	nnsucpred 4717	The successor of the precedessor of a nonzero natural number. (Contributed by Jim Kingdon, 31-Jul-2022.)
⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → suc ∪ 𝐴 = 𝐴)
Theorem	nndceq0 4718	A natural number is either zero or nonzero. Decidable equality for natural numbers is a special case of the law of the excluded middle which holds in most constructive set theories including ours. (Contributed by Jim Kingdon, 5-Jan-2019.)
⊢ (𝐴 ∈ ω → DECID 𝐴 = ∅)
Theorem	0elnn 4719	A natural number is either the empty set or has the empty set as an element. (Contributed by Jim Kingdon, 23-Aug-2019.)
⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
Theorem	nn0eln0 4720	A natural number is nonempty iff it contains the empty set. Although in constructive mathematics it is generally more natural to work with inhabited sets and ignore the whole concept of nonempty sets, in the specific case of natural numbers this theorem may be helpful in converting proofs which were written assuming excluded middle. (Contributed by Jim Kingdon, 28-Aug-2019.)
⊢ (𝐴 ∈ ω → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅))
Theorem	nnregexmid 4721*	If inhabited sets of natural numbers always have minimal elements, excluded middle follows. The argument is essentially the same as regexmid 4635 and the larger lesson is that although natural numbers may behave "non-constructively" even in a constructive set theory (for example see nndceq 6672 or nntri3or 6666), sets of natural numbers are a different animal. (Contributed by Jim Kingdon, 6-Sep-2019.)
⊢ ((𝑥 ⊆ ω ∧ ∃𝑦 𝑦 ∈ 𝑥) → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)))    ⇒   ⊢ (𝜑 ∨ ¬ 𝜑)
Theorem	omsinds 4722*	Strong (or "total") induction principle over ω. (Contributed by Scott Fenton, 17-Jul-2015.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 ∈ ω → (∀𝑦 ∈ 𝑥 𝜓 → 𝜑))    ⇒   ⊢ (𝐴 ∈ ω → 𝜒)
Theorem	nnpredcl 4723	The predecessor of a natural number is a natural number. This theorem is most interesting when the natural number is a successor (as seen in theorems like onsucuni2 4664) but also holds when it is ∅ by uni0 3921. (Contributed by Jim Kingdon, 31-Jul-2022.)
⊢ (𝐴 ∈ ω → ∪ 𝐴 ∈ ω)
Theorem	nnpredlt 4724	The predecessor (see nnpredcl 4723) of a nonzero natural number is less than (see df-iord 4465) that number. (Contributed by Jim Kingdon, 14-Sep-2024.)
⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∪ 𝐴 ∈ 𝐴)
2.6.6  Relations
Syntax	cxp 4725	Extend the definition of a class to include the cross product.
class (𝐴 × 𝐵)
Syntax	ccnv 4726	Extend the definition of a class to include the converse of a class.
class ◡𝐴
Syntax	cdm 4727	Extend the definition of a class to include the domain of a class.
class dom 𝐴
Syntax	crn 4728	Extend the definition of a class to include the range of a class.
class ran 𝐴
Syntax	cres 4729	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 4730	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 4731	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 4732	Extend the definition of a wff to include the relation predicate. (Read: 𝐴 is a relation.)
wff Rel 𝐴
Definition	df-xp 4733*	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⟩}). Another example is that the set of rational numbers is defined using the Cartesian product as (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 4734	Define the relation predicate. Definition 6.4(1) of [TakeutiZaring] p. 23. For alternate definitions, see dfrel2 5189 and dfrel3 5196. (Contributed by NM, 1-Aug-1994.)
⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
Definition	df-cnv 4735*	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 4915 (see df-br 4090 and df-rel 4734 for more on relations). For example, ◡{⟨2, 6⟩, ⟨3, 9⟩} = {⟨6, 2⟩, ⟨9, 3⟩}. We use Quine's breve accent (smile) notation. Like Quine, we use it as a prefix, which eliminates the need for parentheses. "Converse" is Quine's terminology. Some authors use a "minus one" exponent and call it "inverse", especially when the argument is a function, although this is not in general a genuine inverse. (Contributed by NM, 4-Jul-1994.)
⊢ ◡𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
Definition	df-co 4736*	Define the composition of two classes. Definition 6.6(3) of [TakeutiZaring] p. 24. Note that Definition 7 of [Suppes] p. 63 reverses 𝐴 and 𝐵, uses a slash instead of ∘, and calls the operation "relative product". (Contributed by NM, 4-Jul-1994.)
⊢ (𝐴 ∘ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)}
Definition	df-dm 4737*	Define the domain of a class. Definition 3 of [Suppes] p. 59. For example, F = { ⟨ 2 , 6 ⟩, ⟨ 3 , 9 ⟩ } → dom F = { 2 , 3 } . Contrast with range (defined in df-rn 4738). For alternate definitions see dfdm2 5273, dfdm3 4919, and dfdm4 4925. 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 4738	Define the range of a class. For example, F = { ⟨ 2 , 6 ⟩, ⟨ 3 , 9 ⟩ } -> ran F = { 6 , 9 } . Contrast with domain (defined in df-dm 4737). For alternate definitions, see dfrn2 4920, dfrn3 4921, and dfrn4 5199. 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
Definition	df-res 4739	Define the restriction of a class. Definition 6.6(1) of [TakeutiZaring] p. 24. For example, (𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} ∧ 𝐵 = {1, 2}) → (𝐹 ↾ 𝐵) = {⟨2, 6⟩}. 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. (Contributed by NM, 2-Aug-1994.)
⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
Definition	df-ima 4740	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 } . Contrast with restriction (df-res 4739) and range (df-rn 4738). For an alternate definition, see dfima2 5080. (Contributed by NM, 2-Aug-1994.)
⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
Theorem	xpeq1 4741	Equality theorem for cross product. (Contributed by NM, 4-Jul-1994.)
⊢ (𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶))
Theorem	xpeq2 4742	Equality theorem for cross product. (Contributed by NM, 5-Jul-1994.)
⊢ (𝐴 = 𝐵 → (𝐶 × 𝐴) = (𝐶 × 𝐵))
Theorem	elxpi 4743*	Membership in a cross product. Uses fewer axioms than elxp 4744. (Contributed by NM, 4-Jul-1994.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
Theorem	elxp 4744*	Membership in a cross product. (Contributed by NM, 4-Jul-1994.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
Theorem	elxp2 4745*	Membership in a cross product. (Contributed by NM, 23-Feb-2004.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐴 = ⟨𝑥, 𝑦⟩)
Theorem	xpeq12 4746	Equality theorem for cross product. (Contributed by FL, 31-Aug-2009.)
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷))
Theorem	xpeq1i 4747	Equality inference for cross product. (Contributed by NM, 21-Dec-2008.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴 × 𝐶) = (𝐵 × 𝐶)
Theorem	xpeq2i 4748	Equality inference for cross product. (Contributed by NM, 21-Dec-2008.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶 × 𝐴) = (𝐶 × 𝐵)
Theorem	xpeq12i 4749	Equality inference for cross product. (Contributed by FL, 31-Aug-2009.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ (𝐴 × 𝐶) = (𝐵 × 𝐷)
Theorem	xpeq1d 4750	Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 × 𝐶) = (𝐵 × 𝐶))
Theorem	xpeq2d 4751	Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 × 𝐴) = (𝐶 × 𝐵))
Theorem	xpeq12d 4752	Equality deduction for Cartesian product. (Contributed by NM, 8-Dec-2013.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 × 𝐶) = (𝐵 × 𝐷))
Theorem	sqxpeqd 4753	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 4754	Bound-variable hypothesis builder for cross product. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥(𝐴 × 𝐵)
Theorem	0nelxp 4755	The empty set is not a member of a cross product. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ ¬ ∅ ∈ (𝐴 × 𝐵)
Theorem	0nelelxp 4756	A member of a cross product (ordered pair) doesn't contain the empty set. (Contributed by NM, 15-Dec-2008.)
⊢ (𝐶 ∈ (𝐴 × 𝐵) → ¬ ∅ ∈ 𝐶)
Theorem	opelxp 4757	Ordered pair membership in a cross 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	brxp 4758	Binary relation on a cross product. (Contributed by NM, 22-Apr-2004.)
⊢ (𝐴(𝐶 × 𝐷)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷))
Theorem	opelxpi 4759	Ordered pair membership in a cross product (implication). (Contributed by NM, 28-May-1995.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷))
Theorem	opelxpd 4760	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	opelxp1 4761	The first member of an ordered pair of classes in a cross product belongs to first cross product argument. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) → 𝐴 ∈ 𝐶)
Theorem	opelxp2 4762	The second member of an ordered pair of classes in a cross product belongs to second cross product argument. (Contributed by Mario Carneiro, 26-Apr-2015.)
⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) → 𝐵 ∈ 𝐷)
Theorem	otelxp1 4763	The first member of an ordered triple of classes in a cross product belongs to first cross product argument. (Contributed by NM, 28-May-2008.)
⊢ (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ ((𝑅 × 𝑆) × 𝑇) → 𝐴 ∈ 𝑅)
Theorem	opabssxpd 4764*	An ordered-pair class abstraction is a subset of a Cartesian product. Formerly part of proof for opabex2 6362. (Contributed by AV, 26-Nov-2021.)
⊢ ((𝜑 ∧ 𝜓) → 𝑥 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ 𝐵)    ⇒   ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ (𝐴 × 𝐵))
Theorem	rabxp 4765*	Membership in a class builder restricted to a cross product. (Contributed by NM, 20-Feb-2014.)
⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓))    ⇒   ⊢ {𝑥 ∈ (𝐴 × 𝐵) ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ∧ 𝜓)}
Theorem	brrelex12 4766	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 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
Theorem	brrelex1 4767	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 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V)
Theorem	brrelex 4768	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 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V)
Theorem	brrelex2 4769	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 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V)
Theorem	brrelex12i 4770	Two classes that are related by a binary relation are sets. (An artifact of our ordered pair definition.) (Contributed by BJ, 3-Oct-2022.)
⊢ Rel 𝑅    ⇒   ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
Theorem	brrelex1i 4771	The first argument of a binary relation exists. (An artifact of our ordered pair definition.) (Contributed by NM, 4-Jun-1998.)
⊢ Rel 𝑅    ⇒   ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ V)
Theorem	brrelex2i 4772	The second argument of a binary relation exists. (An artifact of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.)
⊢ Rel 𝑅    ⇒   ⊢ (𝐴𝑅𝐵 → 𝐵 ∈ V)
Theorem	nprrel 4773	No proper class is related to anything via any relation. (Contributed by Roy F. Longton, 30-Jul-2005.)
⊢ Rel 𝑅    &   ⊢ ¬ 𝐴 ∈ V    ⇒   ⊢ ¬ 𝐴𝑅𝐵
Theorem	0nelrel 4774	A binary relation does not contain the empty set. (Contributed by AV, 15-Nov-2021.)
⊢ (Rel 𝑅 → ∅ ∉ 𝑅)
Theorem	fconstmpt 4775*	Representation of a constant function using the mapping operation. (Note that 𝑥 cannot appear free in 𝐵.) (Contributed by NM, 12-Oct-1999.) (Revised by Mario Carneiro, 16-Nov-2013.)
⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵)
Theorem	vtoclr 4776*	Variable to class conversion of transitive relation. (Contributed by NM, 9-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ Rel 𝑅    &   ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)    ⇒   ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)
Theorem	opelvvg 4777	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	opelvv 4778	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	opthprc 4779	Justification theorem for an ordered pair definition that works for any classes, including proper classes. This is a possible definition implied by the footnote in [Jech] p. 78, which says, "The sophisticated reader will not object to our use of a pair of classes." (Contributed by NM, 28-Sep-2003.)
⊢ (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
Theorem	brel 4780	Two things in a binary relation belong to the relation's domain. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ 𝑅 ⊆ (𝐶 × 𝐷)    ⇒   ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷))
Theorem	brab2a 4781*	Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 9-Nov-2015.)
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))    &   ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜑)}    ⇒   ⊢ (𝐴𝑅𝐵 ↔ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ 𝜓))
Theorem	elxp3 4782*	Membership in a cross product. (Contributed by NM, 5-Mar-1995.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥∃𝑦(⟨𝑥, 𝑦⟩ = 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶)))
Theorem	opeliunxp 4783	Membership in a union of cross products. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by Mario Carneiro, 1-Jan-2017.)
⊢ (⟨𝑥, 𝐶⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵))
Theorem	xpundi 4784	Distributive law for cross product over union. Theorem 103 of [Suppes] p. 52. (Contributed by NM, 12-Aug-2004.)
⊢ (𝐴 × (𝐵 ∪ 𝐶)) = ((𝐴 × 𝐵) ∪ (𝐴 × 𝐶))
Theorem	xpundir 4785	Distributive law for cross product over union. Similar to Theorem 103 of [Suppes] p. 52. (Contributed by NM, 30-Sep-2002.)
⊢ ((𝐴 ∪ 𝐵) × 𝐶) = ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶))
Theorem	xpiundi 4786*	Distributive law for cross product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.)
⊢ (𝐶 × ∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)
Theorem	xpiundir 4787*	Distributive law for cross product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.)
⊢ (∪ 𝑥 ∈ 𝐴 𝐵 × 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 × 𝐶)
Theorem	iunxpconst 4788*	Membership in a union of cross products when the second factor is constant. (Contributed by Mario Carneiro, 29-Dec-2014.)
⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵)
Theorem	xpun 4789	The cross product of two unions. (Contributed by NM, 12-Aug-2004.)
⊢ ((𝐴 ∪ 𝐵) × (𝐶 ∪ 𝐷)) = (((𝐴 × 𝐶) ∪ (𝐴 × 𝐷)) ∪ ((𝐵 × 𝐶) ∪ (𝐵 × 𝐷)))
Theorem	elvv 4790*	Membership in universal class of ordered pairs. (Contributed by NM, 4-Jul-1994.)
⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
Theorem	elvvv 4791*	Membership in universal class of ordered triples. (Contributed by NM, 17-Dec-2008.)
⊢ (𝐴 ∈ ((V × V) × V) ↔ ∃𝑥∃𝑦∃𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
Theorem	elvvuni 4792	An ordered pair contains its union. (Contributed by NM, 16-Sep-2006.)
⊢ (𝐴 ∈ (V × V) → ∪ 𝐴 ∈ 𝐴)
Theorem	mosubopt 4793*	"At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-Aug-2007.)
⊢ (∀𝑦∀𝑧∃*𝑥𝜑 → ∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑))
Theorem	mosubop 4794*	"At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-May-1995.)
⊢ ∃*𝑥𝜑    ⇒   ⊢ ∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
Theorem	brinxp2 4795	Intersection of binary relation with Cartesian product. (Contributed by NM, 3-Mar-2007.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ (𝐴(𝑅 ∩ (𝐶 × 𝐷))𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴𝑅𝐵))
Theorem	brinxp 4796	Intersection of binary relation with Cartesian product. (Contributed by NM, 9-Mar-1997.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴𝑅𝐵 ↔ 𝐴(𝑅 ∩ (𝐶 × 𝐷))𝐵))
Theorem	poinxp 4797	Intersection of partial order with cross product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)
⊢ (𝑅 Po 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Po 𝐴)
Theorem	soinxp 4798	Intersection of linear order with cross product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)
⊢ (𝑅 Or 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Or 𝐴)
Theorem	seinxp 4799	Intersection of set-like relation with cross product of its field. (Contributed by Mario Carneiro, 22-Jun-2015.)
⊢ (𝑅 Se 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Se 𝐴)
Theorem	posng 4800	Partial ordering of a singleton. (Contributed by Jim Kingdon, 5-Dec-2018.)
⊢ ((Rel 𝑅 ∧ 𝐴 ∈ V) → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴))