P. 43 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 4201-4300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	sspwb 4201	Classes are subclasses if and only if their power classes are subclasses. Exercise 18 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.)
⊢ (𝐴 ⊆ 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵)
Theorem	unipw 4202	A class equals the union of its power class. Exercise 6(a) of [Enderton] p. 38. (Contributed by NM, 14-Oct-1996.) (Proof shortened by Alan Sare, 28-Dec-2008.)
⊢ ∪ 𝒫 𝐴 = 𝐴
Theorem	pwel 4203	Membership of a power class. Exercise 10 of [Enderton] p. 26. (Contributed by NM, 13-Jan-2007.)
⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ 𝒫 𝒫 ∪ 𝐵)
Theorem	pwtr 4204	A class is transitive iff its power class is transitive. (Contributed by Alan Sare, 25-Aug-2011.) (Revised by Mario Carneiro, 15-Jun-2014.)
⊢ (Tr 𝐴 ↔ Tr 𝒫 𝐴)
Theorem	ssextss 4205*	An extensionality-like principle defining subclass in terms of subsets. (Contributed by NM, 30-Jun-2004.)
⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵))
Theorem	ssext 4206*	An extensionality-like principle that uses the subset instead of the membership relation: two classes are equal iff they have the same subsets. (Contributed by NM, 30-Jun-2004.)
⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐵))
Theorem	nssssr 4207*	Negation of subclass relationship. Compare nssr 3207. (Contributed by Jim Kingdon, 17-Sep-2018.)
⊢ (∃𝑥(𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵) → ¬ 𝐴 ⊆ 𝐵)
Theorem	pweqb 4208	Classes are equal if and only if their power classes are equal. Exercise 19 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.)
⊢ (𝐴 = 𝐵 ↔ 𝒫 𝐴 = 𝒫 𝐵)
Theorem	intid 4209*	The intersection of all sets to which a set belongs is the singleton of that set. (Contributed by NM, 5-Jun-2009.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} = {𝐴}
Theorem	euabex 4210	The abstraction of a wff with existential uniqueness exists. (Contributed by NM, 25-Nov-1994.)
⊢ (∃!𝑥𝜑 → {𝑥 ∣ 𝜑} ∈ V)
Theorem	mss 4211*	An inhabited class (even if proper) has an inhabited subset. (Contributed by Jim Kingdon, 17-Sep-2018.)
⊢ (∃𝑦 𝑦 ∈ 𝐴 → ∃𝑥(𝑥 ⊆ 𝐴 ∧ ∃𝑧 𝑧 ∈ 𝑥))
Theorem	exss 4212*	Restricted existence in a class (even if proper) implies restricted existence in a subset. (Contributed by NM, 23-Aug-2003.)
⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑦(𝑦 ⊆ 𝐴 ∧ ∃𝑥 ∈ 𝑦 𝜑))
Theorem	opexg 4213	An ordered pair of sets is a set. (Contributed by Jim Kingdon, 11-Jan-2019.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⟨𝐴, 𝐵⟩ ∈ V)
Theorem	opex 4214	An ordered pair of sets is a set. (Contributed by Jim Kingdon, 24-Sep-2018.) (Revised by Mario Carneiro, 24-May-2019.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ⟨𝐴, 𝐵⟩ ∈ V
Theorem	otexg 4215	An ordered triple of sets is a set. (Contributed by Jim Kingdon, 19-Sep-2018.)
⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ⟨𝐴, 𝐵, 𝐶⟩ ∈ V)
Theorem	elop 4216	An ordered pair has two elements. Exercise 3 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ (𝐴 ∈ ⟨𝐵, 𝐶⟩ ↔ (𝐴 = {𝐵} ∨ 𝐴 = {𝐵, 𝐶}))
Theorem	opi1 4217	One of the two elements in an ordered pair. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ {𝐴} ∈ ⟨𝐴, 𝐵⟩
Theorem	opi2 4218	One of the two elements of an ordered pair. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ {𝐴, 𝐵} ∈ ⟨𝐴, 𝐵⟩
2.3.4  Ordered pair theorem
Theorem	opm 4219*	An ordered pair is inhabited iff the arguments are sets. (Contributed by Jim Kingdon, 21-Sep-2018.)
⊢ (∃𝑥 𝑥 ∈ ⟨𝐴, 𝐵⟩ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
Theorem	opnzi 4220	An ordered pair is nonempty if the arguments are sets (it is also inhabited; see opm 4219). (Contributed by Mario Carneiro, 26-Apr-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ⟨𝐴, 𝐵⟩ ≠ ∅
Theorem	opth1 4221	Equality of the first members of equal ordered pairs. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐴 = 𝐶)
Theorem	opth 4222	The ordered pair theorem. If two ordered pairs are equal, their first elements are equal and their second elements are equal. Exercise 6 of [TakeutiZaring] p. 16. Note that 𝐶 and 𝐷 are not required to be sets due our specific ordered pair definition. (Contributed by NM, 28-May-1995.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
Theorem	opthg 4223	Ordered pair theorem. 𝐶 and 𝐷 are not required to be sets under our specific ordered pair definition. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
Theorem	opthg2 4224	Ordered pair theorem. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ ((𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
Theorem	opth2 4225	Ordered pair theorem. (Contributed by NM, 21-Sep-2014.)
⊢ 𝐶 ∈ V    &   ⊢ 𝐷 ∈ V    ⇒   ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
Theorem	otth2 4226	Ordered triple theorem, with triple express with ordered pairs. (Contributed by NM, 1-May-1995.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝑅 ∈ V    ⇒   ⊢ (⟨⟨𝐴, 𝐵⟩, 𝑅⟩ = ⟨⟨𝐶, 𝐷⟩, 𝑆⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ∧ 𝑅 = 𝑆))
Theorem	otth 4227	Ordered triple theorem. (Contributed by NM, 25-Sep-2014.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝑅 ∈ V    ⇒   ⊢ (⟨𝐴, 𝐵, 𝑅⟩ = ⟨𝐶, 𝐷, 𝑆⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ∧ 𝑅 = 𝑆))
Theorem	eqvinop 4228*	A variable introduction law for ordered pairs. Analog of Lemma 15 of [Monk2] p. 109. (Contributed by NM, 28-May-1995.)
⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ (𝐴 = ⟨𝐵, 𝐶⟩ ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩))
Theorem	copsexg 4229*	Substitution of class 𝐴 for ordered pair ⟨𝑥, 𝑦⟩. (Contributed by NM, 27-Dec-1996.) (Revised by Andrew Salmon, 11-Jul-2011.)
⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
Theorem	copsex2t 4230*	Closed theorem form of copsex2g 4231. (Contributed by NM, 17-Feb-2013.)
⊢ ((∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓))
Theorem	copsex2g 4231*	Implicit substitution inference for ordered pairs. (Contributed by NM, 28-May-1995.)
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓))
Theorem	copsex4g 4232*	An implicit substitution inference for 2 ordered pairs. (Contributed by NM, 5-Aug-1995.)
⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆)) → (∃𝑥∃𝑦∃𝑧∃𝑤((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑧, 𝑤⟩) ∧ 𝜑) ↔ 𝜓))
Theorem	0nelop 4233	A property of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.)
⊢ ¬ ∅ ∈ ⟨𝐴, 𝐵⟩
Theorem	opeqex 4234	Equivalence of existence implied by equality of ordered pairs. (Contributed by NM, 28-May-2008.)
⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐶 ∈ V ∧ 𝐷 ∈ V)))
Theorem	opcom 4235	An ordered pair commutes iff its members are equal. (Contributed by NM, 28-May-2009.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐵, 𝐴⟩ ↔ 𝐴 = 𝐵)
Theorem	moop2 4236*	"At most one" property of an ordered pair. (Contributed by NM, 11-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ 𝐵 ∈ V    ⇒   ⊢ ∃*𝑥 𝐴 = ⟨𝐵, 𝑥⟩
Theorem	opeqsn 4237	Equivalence for an ordered pair equal to a singleton. (Contributed by NM, 3-Jun-2008.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ (⟨𝐴, 𝐵⟩ = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴}))
Theorem	opeqpr 4238	Equivalence for an ordered pair equal to an unordered pair. (Contributed by NM, 3-Jun-2008.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    &   ⊢ 𝐷 ∈ V    ⇒   ⊢ (⟨𝐴, 𝐵⟩ = {𝐶, 𝐷} ↔ ((𝐶 = {𝐴} ∧ 𝐷 = {𝐴, 𝐵}) ∨ (𝐶 = {𝐴, 𝐵} ∧ 𝐷 = {𝐴})))
Theorem	euotd 4239*	Prove existential uniqueness for an ordered triple. (Contributed by Mario Carneiro, 20-May-2015.)
⊢ (𝜑 → 𝐴 ∈ V)    &   ⊢ (𝜑 → 𝐵 ∈ V)    &   ⊢ (𝜑 → 𝐶 ∈ V)    &   ⊢ (𝜑 → (𝜓 ↔ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ∧ 𝑐 = 𝐶)))    ⇒   ⊢ (𝜑 → ∃!𝑥∃𝑎∃𝑏∃𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
Theorem	uniop 4240	The union of an ordered pair. Theorem 65 of [Suppes] p. 39. (Contributed by NM, 17-Aug-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ∪ ⟨𝐴, 𝐵⟩ = {𝐴, 𝐵}
Theorem	uniopel 4241	Ordered pair membership is inherited by class union. (Contributed by NM, 13-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → ∪ ⟨𝐴, 𝐵⟩ ∈ ∪ 𝐶)
2.3.5  Ordered-pair class abstractions (cont.)
Theorem	opabid 4242	The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
Theorem	elopab 4243*	Membership in a class abstraction of ordered pairs. (Contributed by NM, 24-Mar-1998.)
⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
Theorem	opelopabsbALT 4244*	The law of concretion in terms of substitutions. Less general than opelopabsb 4245, but having a much shorter proof. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)
Theorem	opelopabsb 4245*	The law of concretion in terms of substitutions. (Contributed by NM, 30-Sep-2002.) (Revised by Mario Carneiro, 18-Nov-2016.)
⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑)
Theorem	brabsb 4246*	The law of concretion in terms of substitutions. (Contributed by NM, 17-Mar-2008.)
⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}    ⇒   ⊢ (𝐴𝑅𝐵 ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑)
Theorem	opelopabt 4247*	Closed theorem form of opelopab 4256. (Contributed by NM, 19-Feb-2013.)
⊢ ((∀𝑥∀𝑦(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∀𝑥∀𝑦(𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))
Theorem	opelopabga 4248*	The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 19-Dec-2013.)
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜓))
Theorem	brabga 4249*	The law of concretion for a binary relation. (Contributed by Mario Carneiro, 19-Dec-2013.)
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))    &   ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴𝑅𝐵 ↔ 𝜓))
Theorem	opelopab2a 4250*	Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 19-Dec-2013.)
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜑)} ↔ 𝜓))
Theorem	opelopaba 4251*	The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 19-Dec-2013.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜓)
Theorem	braba 4252*	The law of concretion for a binary relation. (Contributed by NM, 19-Dec-2013.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))    &   ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}    ⇒   ⊢ (𝐴𝑅𝐵 ↔ 𝜓)
Theorem	opelopabg 4253*	The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 19-Dec-2013.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))
Theorem	brabg 4254*	The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 19-Dec-2013.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    &   ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}    ⇒   ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴𝑅𝐵 ↔ 𝜒))
Theorem	opelopab2 4255*	Ordered pair membership in an ordered pair class abstraction. (Contributed by NM, 14-Oct-2007.) (Revised by Mario Carneiro, 19-Dec-2013.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    ⇒   ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜑)} ↔ 𝜒))
Theorem	opelopab 4256*	The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 16-May-1995.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒)
Theorem	brab 4257*	The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    &   ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}    ⇒   ⊢ (𝐴𝑅𝐵 ↔ 𝜒)
Theorem	opelopabaf 4258*	The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4256 uses bound-variable hypotheses in place of distinct variable conditions. (Contributed by Mario Carneiro, 19-Dec-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
⊢ Ⅎ𝑥𝜓    &   ⊢ Ⅎ𝑦𝜓    &   ⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜓)
Theorem	opelopabf 4259*	The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4256 uses bound-variable hypotheses in place of distinct variable conditions. (Contributed by NM, 19-Dec-2008.)
⊢ Ⅎ𝑥𝜓    &   ⊢ Ⅎ𝑦𝜒    &   ⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒)
Theorem	ssopab2 4260	Equivalence of ordered pair abstraction subclass and implication. (Contributed by NM, 27-Dec-1996.) (Revised by Mario Carneiro, 19-May-2013.)
⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓})
Theorem	ssopab2b 4261	Equivalence of ordered pair abstraction subclass and implication. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ∀𝑥∀𝑦(𝜑 → 𝜓))
Theorem	ssopab2i 4262	Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 5-Apr-1995.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓}
Theorem	ssopab2dv 4263*	Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 19-Jan-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜒})
Theorem	eqopab2b 4264	Equivalence of ordered pair abstraction equality and biconditional. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ∀𝑥∀𝑦(𝜑 ↔ 𝜓))
Theorem	opabm 4265*	Inhabited ordered pair class abstraction. (Contributed by Jim Kingdon, 29-Sep-2018.)
⊢ (∃𝑧 𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦𝜑)
Theorem	iunopab 4266*	Move indexed union inside an ordered-pair abstraction. (Contributed by Stefan O'Rear, 20-Feb-2015.)
⊢ ∪ 𝑧 ∈ 𝐴 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 𝜑}
2.3.6  Power class of union and intersection
Theorem	pwin 4267	The power class of the intersection of two classes is the intersection of their power classes. Exercise 4.12(j) of [Mendelson] p. 235. (Contributed by NM, 23-Nov-2003.)
⊢ 𝒫 (𝐴 ∩ 𝐵) = (𝒫 𝐴 ∩ 𝒫 𝐵)
Theorem	pwunss 4268	The power class of the union of two classes includes the union of their power classes. Exercise 4.12(k) of [Mendelson] p. 235. (Contributed by NM, 23-Nov-2003.)
⊢ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵)
Theorem	pwssunim 4269	The power class of the union of two classes is a subset of the union of their power classes, if one class is a subclass of the other. One direction of Exercise 4.12(l) of [Mendelson] p. 235. (Contributed by Jim Kingdon, 30-Sep-2018.)
⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) → 𝒫 (𝐴 ∪ 𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵))
Theorem	pwundifss 4270	Break up the power class of a union into a union of smaller classes. (Contributed by Jim Kingdon, 30-Sep-2018.)
⊢ ((𝒫 (𝐴 ∪ 𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴) ⊆ 𝒫 (𝐴 ∪ 𝐵)
Theorem	pwunim 4271	The power class of the union of two classes equals the union of their power classes, iff one class is a subclass of the other. Part of Exercise 7(b) of [Enderton] p. 28. (Contributed by Jim Kingdon, 30-Sep-2018.)
⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) → 𝒫 (𝐴 ∪ 𝐵) = (𝒫 𝐴 ∪ 𝒫 𝐵))
2.3.7  Epsilon and identity relations
Syntax	cep 4272	Extend class notation to include the epsilon relation.
class E
Syntax	cid 4273	Extend the definition of a class to include identity relation.
class I
Definition	df-eprel 4274*	Define the epsilon relation. Similar to Definition 6.22 of [TakeutiZaring] p. 30. The epsilon relation and set membership are the same, that is, (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵) when 𝐵 is a set by epelg 4275. Thus, 5 E { 1 , 5 }. (Contributed by NM, 13-Aug-1995.)
⊢ E = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦}
Theorem	epelg 4275	The epsilon relation and membership are the same. General version of epel 4277. (Contributed by Scott Fenton, 27-Mar-2011.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))
Theorem	epelc 4276	The epsilon relationship and the membership relation are the same. (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)
Theorem	epel 4277	The epsilon relation and the membership relation are the same. (Contributed by NM, 13-Aug-1995.)
⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
Definition	df-id 4278*	Define the identity relation. Definition 9.15 of [Quine] p. 64. For example, 5 I 5 and ¬ 4 I 5. (Contributed by NM, 13-Aug-1995.)
⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
2.3.8  Partial and total orderings We have not yet defined relations (df-rel 4618), but here we introduce a few related notions we will use to develop ordinals. The class variable 𝑅 is no different from other class variables, but it reminds us that typically it represents what we will later call a "relation".
Syntax	wpo 4279	Extend wff notation to include the strict partial ordering predicate. Read: ' 𝑅 is a partial order on 𝐴.'
wff 𝑅 Po 𝐴
Syntax	wor 4280	Extend wff notation to include the strict linear ordering predicate. Read: ' 𝑅 orders 𝐴.'
wff 𝑅 Or 𝐴
Definition	df-po 4281*	Define the strict partial order predicate. Definition of [Enderton] p. 168. The expression 𝑅 Po 𝐴 means 𝑅 is a partial order on 𝐴. (Contributed by NM, 16-Mar-1997.)
⊢ (𝑅 Po 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
Definition	df-iso 4282*	Define the strict linear order predicate. The expression 𝑅 Or 𝐴 is true if relationship 𝑅 orders 𝐴. The property 𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦) is called weak linearity by Proposition 11.2.3 of [HoTT], p. (varies). If we assumed excluded middle, it would be equivalent to trichotomy, 𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥. (Contributed by NM, 21-Jan-1996.) (Revised by Jim Kingdon, 4-Oct-2018.)
⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))))
Theorem	poss 4283	Subset theorem for the partial ordering predicate. (Contributed by NM, 27-Mar-1997.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
⊢ (𝐴 ⊆ 𝐵 → (𝑅 Po 𝐵 → 𝑅 Po 𝐴))
Theorem	poeq1 4284	Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.)
⊢ (𝑅 = 𝑆 → (𝑅 Po 𝐴 ↔ 𝑆 Po 𝐴))
Theorem	poeq2 4285	Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.)
⊢ (𝐴 = 𝐵 → (𝑅 Po 𝐴 ↔ 𝑅 Po 𝐵))
Theorem	nfpo 4286	Bound-variable hypothesis builder for partial orders. (Contributed by Stefan O'Rear, 20-Jan-2015.)
⊢ Ⅎ𝑥𝑅    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥 𝑅 Po 𝐴
Theorem	nfso 4287	Bound-variable hypothesis builder for total orders. (Contributed by Stefan O'Rear, 20-Jan-2015.)
⊢ Ⅎ𝑥𝑅    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥 𝑅 Or 𝐴
Theorem	pocl 4288	Properties of partial order relation in class notation. (Contributed by NM, 27-Mar-1997.)
⊢ (𝑅 Po 𝐴 → ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷))))
Theorem	ispod 4289*	Sufficient conditions for a partial order. (Contributed by NM, 9-Jul-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥𝑅𝑥)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))    ⇒   ⊢ (𝜑 → 𝑅 Po 𝐴)
Theorem	swopolem 4290*	Perform the substitutions into the strict weak ordering law. (Contributed by Mario Carneiro, 31-Dec-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)))    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) → (𝑋𝑅𝑌 → (𝑋𝑅𝑍 ∨ 𝑍𝑅𝑌)))
Theorem	swopo 4291*	A strict weak order is a partial order. (Contributed by Mario Carneiro, 9-Jul-2014.)
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑦𝑅𝑧 → ¬ 𝑧𝑅𝑦))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)))    ⇒   ⊢ (𝜑 → 𝑅 Po 𝐴)
Theorem	poirr 4292	A partial order relation is irreflexive. (Contributed by NM, 27-Mar-1997.)
⊢ ((𝑅 Po 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵)
Theorem	potr 4293	A partial order relation is a transitive relation. (Contributed by NM, 27-Mar-1997.)
⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷))
Theorem	po2nr 4294	A partial order relation has no 2-cycle loops. (Contributed by NM, 27-Mar-1997.)
⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵))
Theorem	po3nr 4295	A partial order relation has no 3-cycle loops. (Contributed by NM, 27-Mar-1997.)
⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷 ∧ 𝐷𝑅𝐵))
Theorem	po0 4296	Any relation is a partial ordering of the empty set. (Contributed by NM, 28-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
⊢ 𝑅 Po ∅
Theorem	pofun 4297*	A function preserves a partial order relation. (Contributed by Jeff Madsen, 18-Jun-2011.)
⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌}    &   ⊢ (𝑥 = 𝑦 → 𝑋 = 𝑌)    ⇒   ⊢ ((𝑅 Po 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑋 ∈ 𝐵) → 𝑆 Po 𝐴)
Theorem	sopo 4298	A strict linear order is a strict partial order. (Contributed by NM, 28-Mar-1997.)
⊢ (𝑅 Or 𝐴 → 𝑅 Po 𝐴)
Theorem	soss 4299	Subset theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
⊢ (𝐴 ⊆ 𝐵 → (𝑅 Or 𝐵 → 𝑅 Or 𝐴))
Theorem	soeq1 4300	Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.)
⊢ (𝑅 = 𝑆 → (𝑅 Or 𝐴 ↔ 𝑆 Or 𝐴))