P. 47 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 4601-4700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nnpredlt 4601	The predecessor (see nnpredcl 4600) of a nonzero natural number is less than (see df-iord 4344) that number. (Contributed by Jim Kingdon, 14-Sep-2024.)
⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∪ 𝐴 ∈ 𝐴)
2.6.6  Relations
Syntax	cxp 4602	Extend the definition of a class to include the cross product.
class (𝐴 × 𝐵)
Syntax	ccnv 4603	Extend the definition of a class to include the converse of a class.
class ◡𝐴
Syntax	cdm 4604	Extend the definition of a class to include the domain of a class.
class dom 𝐴
Syntax	crn 4605	Extend the definition of a class to include the range of a class.
class ran 𝐴
Syntax	cres 4606	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 4607	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 4608	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 4609	Extend the definition of a wff to include the relation predicate. (Read: 𝐴 is a relation.)
wff Rel 𝐴
Definition	df-xp 4610*	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 4611	Define the relation predicate. Definition 6.4(1) of [TakeutiZaring] p. 23. For alternate definitions, see dfrel2 5054 and dfrel3 5061. (Contributed by NM, 1-Aug-1994.)
⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
Definition	df-cnv 4612*	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 4787 (see df-br 3983 and df-rel 4611 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 4613*	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 4614*	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 4615). For alternate definitions see dfdm2 5138, dfdm3 4791, and dfdm4 4796. 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 4615	Define the range of a class. For example, F = { ⟨ 2 , 6 ⟩, ⟨ 3 , 9 ⟩ } -> ran F = { 6 , 9 } . Contrast with domain (defined in df-dm 4614). For alternate definitions, see dfrn2 4792, dfrn3 4793, and dfrn4 5064. 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 4616	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 4617	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 4616) and range (df-rn 4615). For an alternate definition, see dfima2 4948. (Contributed by NM, 2-Aug-1994.)
⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
Theorem	xpeq1 4618	Equality theorem for cross product. (Contributed by NM, 4-Jul-1994.)
⊢ (𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶))
Theorem	xpeq2 4619	Equality theorem for cross product. (Contributed by NM, 5-Jul-1994.)
⊢ (𝐴 = 𝐵 → (𝐶 × 𝐴) = (𝐶 × 𝐵))
Theorem	elxpi 4620*	Membership in a cross product. Uses fewer axioms than elxp 4621. (Contributed by NM, 4-Jul-1994.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
Theorem	elxp 4621*	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 4622*	Membership in a cross product. (Contributed by NM, 23-Feb-2004.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐴 = ⟨𝑥, 𝑦⟩)
Theorem	xpeq12 4623	Equality theorem for cross product. (Contributed by FL, 31-Aug-2009.)
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷))
Theorem	xpeq1i 4624	Equality inference for cross product. (Contributed by NM, 21-Dec-2008.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴 × 𝐶) = (𝐵 × 𝐶)
Theorem	xpeq2i 4625	Equality inference for cross product. (Contributed by NM, 21-Dec-2008.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶 × 𝐴) = (𝐶 × 𝐵)
Theorem	xpeq12i 4626	Equality inference for cross product. (Contributed by FL, 31-Aug-2009.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ (𝐴 × 𝐶) = (𝐵 × 𝐷)
Theorem	xpeq1d 4627	Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 × 𝐶) = (𝐵 × 𝐶))
Theorem	xpeq2d 4628	Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 × 𝐴) = (𝐶 × 𝐵))
Theorem	xpeq12d 4629	Equality deduction for Cartesian product. (Contributed by NM, 8-Dec-2013.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 × 𝐶) = (𝐵 × 𝐷))
Theorem	sqxpeqd 4630	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 4631	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 4632	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 4633	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 4634	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 4635	Binary relation on a cross product. (Contributed by NM, 22-Apr-2004.)
⊢ (𝐴(𝐶 × 𝐷)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷))
Theorem	opelxpi 4636	Ordered pair membership in a cross product (implication). (Contributed by NM, 28-May-1995.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷))
Theorem	opelxpd 4637	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 4638	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 4639	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 4640	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	rabxp 4641*	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 4642	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 4643	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 4644	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 4645	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 4646	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 4647	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 4648	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 4649	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 4650	A binary relation does not contain the empty set. (Contributed by AV, 15-Nov-2021.)
⊢ (Rel 𝑅 → ∅ ∉ 𝑅)
Theorem	fconstmpt 4651*	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 4652*	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 4653	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 4654	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 4655	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 4656	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 4657*	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 4658*	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 4659	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 4660	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 4661	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 4662*	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 4663*	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 4664*	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 4665	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 4666*	Membership in universal class of ordered pairs. (Contributed by NM, 4-Jul-1994.)
⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
Theorem	elvvv 4667*	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 4668	An ordered pair contains its union. (Contributed by NM, 16-Sep-2006.)
⊢ (𝐴 ∈ (V × V) → ∪ 𝐴 ∈ 𝐴)
Theorem	mosubopt 4669*	"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 4670*	"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 4671	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 4672	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 4673	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 4674	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 4675	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 4676	Partial ordering of a singleton. (Contributed by Jim Kingdon, 5-Dec-2018.)
⊢ ((Rel 𝑅 ∧ 𝐴 ∈ V) → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴))
Theorem	sosng 4677	Strict linear ordering on a singleton. (Contributed by Jim Kingdon, 5-Dec-2018.)
⊢ ((Rel 𝑅 ∧ 𝐴 ∈ V) → (𝑅 Or {𝐴} ↔ ¬ 𝐴𝑅𝐴))
Theorem	opabssxp 4678*	An abstraction relation is a subset of a related cross product. (Contributed by NM, 16-Jul-1995.)
⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)} ⊆ (𝐴 × 𝐵)
Theorem	brab2ga 4679*	The law of concretion for a binary relation. See brab2a 4657 for alternate proof. TODO: should one of them be deleted? (Contributed by Mario Carneiro, 28-Apr-2015.) (Proof modification is discouraged.)
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))    &   ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜑)}    ⇒   ⊢ (𝐴𝑅𝐵 ↔ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ 𝜓))
Theorem	optocl 4680*	Implicit substitution of class for ordered pair. (Contributed by NM, 5-Mar-1995.)
⊢ 𝐷 = (𝐵 × 𝐶)    &   ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → 𝜑)    ⇒   ⊢ (𝐴 ∈ 𝐷 → 𝜓)
Theorem	2optocl 4681*	Implicit substitution of classes for ordered pairs. (Contributed by NM, 12-Mar-1995.)
⊢ 𝑅 = (𝐶 × 𝐷)    &   ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (⟨𝑧, 𝑤⟩ = 𝐵 → (𝜓 ↔ 𝜒))    &   ⊢ (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ (𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷)) → 𝜑)    ⇒   ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑅) → 𝜒)
Theorem	3optocl 4682*	Implicit substitution of classes for ordered pairs. (Contributed by NM, 12-Mar-1995.)
⊢ 𝑅 = (𝐷 × 𝐹)    &   ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (⟨𝑧, 𝑤⟩ = 𝐵 → (𝜓 ↔ 𝜒))    &   ⊢ (⟨𝑣, 𝑢⟩ = 𝐶 → (𝜒 ↔ 𝜃))    &   ⊢ (((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐹) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐹) ∧ (𝑣 ∈ 𝐷 ∧ 𝑢 ∈ 𝐹)) → 𝜑)    ⇒   ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑅) → 𝜃)
Theorem	opbrop 4683*	Ordered pair membership in a relation. Special case. (Contributed by NM, 5-Aug-1995.)
⊢ (((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) ∧ (𝑣 = 𝐶 ∧ 𝑢 = 𝐷)) → (𝜑 ↔ 𝜓))    &   ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ 𝜑))}    ⇒   ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (⟨𝐴, 𝐵⟩𝑅⟨𝐶, 𝐷⟩ ↔ 𝜓))
Theorem	0xp 4684	The cross product with the empty set is empty. Part of Theorem 3.13(ii) of [Monk1] p. 37. (Contributed by NM, 4-Jul-1994.)
⊢ (∅ × 𝐴) = ∅
Theorem	csbxpg 4685	Distribute proper substitution through the cross product of two classes. (Contributed by Alan Sare, 10-Nov-2012.)
⊢ (𝐴 ∈ 𝐷 → ⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 × ⦋𝐴 / 𝑥⦌𝐶))
Theorem	releq 4686	Equality theorem for the relation predicate. (Contributed by NM, 1-Aug-1994.)
⊢ (𝐴 = 𝐵 → (Rel 𝐴 ↔ Rel 𝐵))
Theorem	releqi 4687	Equality inference for the relation predicate. (Contributed by NM, 8-Dec-2006.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (Rel 𝐴 ↔ Rel 𝐵)
Theorem	releqd 4688	Equality deduction for the relation predicate. (Contributed by NM, 8-Mar-2014.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (Rel 𝐴 ↔ Rel 𝐵))
Theorem	nfrel 4689	Bound-variable hypothesis builder for a relation. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥Rel 𝐴
Theorem	sbcrel 4690	Distribute proper substitution through a relation predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Rel 𝑅 ↔ Rel ⦋𝐴 / 𝑥⦌𝑅))
Theorem	relss 4691	Subclass theorem for relation predicate. Theorem 2 of [Suppes] p. 58. (Contributed by NM, 15-Aug-1994.)
⊢ (𝐴 ⊆ 𝐵 → (Rel 𝐵 → Rel 𝐴))
Theorem	ssrel 4692*	A subclass relationship depends only on a relation's ordered pairs. Theorem 3.2(i) of [Monk1] p. 33. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
Theorem	eqrel 4693*	Extensionality principle for relations. Theorem 3.2(ii) of [Monk1] p. 33. (Contributed by NM, 2-Aug-1994.)
⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
Theorem	ssrel2 4694*	A subclass relationship depends only on a relation's ordered pairs. This version of ssrel 4692 is restricted to the relation's domain. (Contributed by Thierry Arnoux, 25-Jan-2018.)
⊢ (𝑅 ⊆ (𝐴 × 𝐵) → (𝑅 ⊆ 𝑆 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆)))
Theorem	relssi 4695*	Inference from subclass principle for relations. (Contributed by NM, 31-Mar-1998.)
⊢ Rel 𝐴    &   ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)    ⇒   ⊢ 𝐴 ⊆ 𝐵
Theorem	relssdv 4696*	Deduction from subclass principle for relations. (Contributed by NM, 11-Sep-2004.)
⊢ (𝜑 → Rel 𝐴)    &   ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
Theorem	eqrelriv 4697*	Inference from extensionality principle for relations. (Contributed by FL, 15-Oct-2012.)
⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)    ⇒   ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → 𝐴 = 𝐵)
Theorem	eqrelriiv 4698*	Inference from extensionality principle for relations. (Contributed by NM, 17-Mar-1995.)
⊢ Rel 𝐴    &   ⊢ Rel 𝐵    &   ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)    ⇒   ⊢ 𝐴 = 𝐵
Theorem	eqbrriv 4699*	Inference from extensionality principle for relations. (Contributed by NM, 12-Dec-2006.)
⊢ Rel 𝐴    &   ⊢ Rel 𝐵    &   ⊢ (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦)    ⇒   ⊢ 𝐴 = 𝐵
Theorem	eqrelrdv 4700*	Deduce equality of relations from equivalence of membership. (Contributed by Rodolfo Medina, 10-Oct-2010.)
⊢ Rel 𝐴    &   ⊢ Rel 𝐵    &   ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)