P. 503 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 50201-50300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	tposresxp 50201	The transposition restricted to a Cartesian product. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ (tpos 𝐹 ↾ (𝐴 × 𝐵)) = tpos (𝐹 ↾ (𝐵 × 𝐴))
Theorem	tposf1o 50202	Condition of a bijective transposition. (Contributed by Zhi Wang, 5-Oct-2025.)
⊢ (𝐹:(𝐴 × 𝐵)⟶1-1onto𝐶 → tpos 𝐹:(𝐵 × 𝐴)⟶1-1onto𝐶)
Theorem	tposid 50203	Swap an ordered pair. (Contributed by Zhi Wang, 5-Oct-2025.)
⊢ (𝑋tpos I 𝑌) = ⟨𝑌, 𝑋⟩
Theorem	tposidres 50204	Swap an ordered pair. (Contributed by Zhi Wang, 5-Oct-2025.)
⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑌tpos ( I ↾ (𝐴 × 𝐵))𝑋) = ⟨𝑋, 𝑌⟩)
Theorem	tposidf1o 50205	The swap function, or the twisting map, is bijective. (Contributed by Zhi Wang, 5-Oct-2025.)
⊢ tpos ( I ↾ (𝐴 × 𝐵)):(𝐵 × 𝐴)⟶1-1onto(𝐴 × 𝐵)
Theorem	tposideq 50206*	Two ways of expressing the swap function. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ (Rel 𝑅 → (tpos I ↾ 𝑅) = (𝑥 ∈ 𝑅 ↦ ∪ ◡{𝑥}))
Theorem	tposideq2 50207*	Two ways of expressing the swap function. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ 𝑅 = (𝐴 × 𝐵)    ⇒   ⊢ (tpos I ↾ 𝑅) = (𝑥 ∈ 𝑅 ↦ ∪ ◡{𝑥})
21.53.6.5  Infinite Cartesian products
Theorem	ixpv 50208*	Infinite Cartesian product of the universal class is the set of functions with a fixed domain. (Contributed by Zhi Wang, 1-Nov-2025.)
⊢ X𝑥 ∈ 𝐴 V = {𝑓 ∣ 𝑓 Fn 𝐴}
Theorem	elixpmpt 50209*	Membership in an indexed Cartesian product over a set is equivalent to being its exact function graph with components of the indicated types. (Contributed by Zhi Wang, 14-Sep-2026.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))
21.53.6.6  Equinumerosity
Theorem	fvconst0ci 50210	A constant function's value is either the constant or the empty set. (An artifact of our function value definition.) (Contributed by Zhi Wang, 18-Sep-2024.)
⊢ 𝐵 ∈ V    &   ⊢ 𝑌 = ((𝐴 × {𝐵})‘𝑋)    ⇒   ⊢ (𝑌 = ∅ ∨ 𝑌 = 𝐵)
Theorem	fvconstdomi 50211	A constant function's value is dominated by the constant. (An artifact of our function value definition.) (Contributed by Zhi Wang, 18-Sep-2024.)
⊢ 𝐵 ∈ V    ⇒   ⊢ ((𝐴 × {𝐵})‘𝑋) ≼ 𝐵
Theorem	f1omo 50212*	There is at most one element in the function value of a constant function whose output is 1o. (An artifact of our function value definition.) Proof could be significantly shortened by fvconstdomi 50211 assuming ax-un 7775 (see f1omoALT 50214). (Contributed by Zhi Wang, 19-Sep-2024.) (Proof shortened by SN, 24-Nov-2025.)
⊢ (𝜑 → 𝐹 = (𝐴 × {1o}))    ⇒   ⊢ (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋))
Theorem	f1omoOLD 50213*	Obsolete version of f1omo 50212 as of 24-Nov-2025. (Contributed by Zhi Wang, 19-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐹 = (𝐴 × {1o}))    ⇒   ⊢ (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋))
Theorem	f1omoALT 50214*	There is at most one element in the function value of a constant function whose output is 1o. (An artifact of our function value definition.) Use f1omo 50212 without assuming ax-un 7775. (Contributed by Zhi Wang, 18-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐹 = (𝐴 × {1o}))    ⇒   ⊢ (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋))
21.53.7  Order sets
21.53.7.1  Real number intervals
Theorem	iccin 50215	Intersection of two closed intervals of extended reals. (Contributed by Zhi Wang, 9-Sep-2024.)
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) → ((𝐴[,]𝐵) ∩ (𝐶[,]𝐷)) = (if(𝐴 ≤ 𝐶, 𝐶, 𝐴)[,]if(𝐵 ≤ 𝐷, 𝐵, 𝐷)))
Theorem	iccdisj2 50216	If the upper bound of one closed interval is less than the lower bound of the other, the intervals are disjoint. (Contributed by Zhi Wang, 9-Sep-2024.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐷 ∈ ℝ* ∧ 𝐵 < 𝐶) → ((𝐴[,]𝐵) ∩ (𝐶[,]𝐷)) = ∅)
Theorem	iccdisj 50217	If the upper bound of one closed interval is less than the lower bound of the other, the intervals are disjoint. (Contributed by Zhi Wang, 9-Sep-2024.)
⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) ∧ 𝐵 < 𝐶) → ((𝐴[,]𝐵) ∩ (𝐶[,]𝐷)) = ∅)
21.53.8  Extensible structures
21.53.8.1  Basic definitions
Theorem	slotresfo 50218*	The condition of a structure component extractor restricted to a class being a surjection. This combined with fonex 50183 can be used to prove a class being proper. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ 𝐸 Fn V    &   ⊢ (𝑘 ∈ 𝐴 → (𝐸‘𝑘) ∈ 𝑉)    &   ⊢ (𝑏 ∈ 𝑉 → 𝐾 ∈ 𝐴)    &   ⊢ (𝑏 ∈ 𝑉 → 𝑏 = (𝐸‘𝐾))    ⇒   ⊢ (𝐸 ↾ 𝐴):𝐴⟶onto𝑉
21.53.9  Moore spaces
Theorem	mreuniss 50219	The union of a collection of closed sets is a subset. (Contributed by Zhi Wang, 29-Sep-2024.)
⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) → ∪ 𝑆 ⊆ 𝑋)
21.53.10  Topology Additional contents for topology.
21.53.10.1  Closure and interior
Theorem	clduni 50220	The union of closed sets is the underlying set of the topology (the union of open sets). (Contributed by Zhi Wang, 6-Sep-2024.)
⊢ (𝐽 ∈ Top → ∪ (Clsd‘𝐽) = ∪ 𝐽)
Theorem	opncldbid 50221*	Conditions on open sets are equivalent to conditions on closed sets. (Contributed by Zhi Wang, 30-Aug-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ ((𝜑 ∧ 𝑥 = (∪ 𝐽 ∖ 𝑦)) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐽 𝜓 ↔ ∀𝑦 ∈ (Clsd‘𝐽)𝜒))
Theorem	opndisj 50222	Two ways of saying that two open sets are disjoint, if 𝐽 is a topology and 𝑋 is an open set. (Contributed by Zhi Wang, 6-Sep-2024.)
⊢ (𝑍 = (∪ 𝐽 ∖ 𝑋) → (𝑌 ∈ (𝐽 ∩ 𝒫 𝑍) ↔ (𝑌 ∈ 𝐽 ∧ (𝑋 ∩ 𝑌) = ∅)))
Theorem	clddisj 50223	Two ways of saying that two closed sets are disjoint, if 𝐽 is a topology and 𝑋 is a closed set. An alternative proof is similar to that of opndisj 50222 with elssuni 4902 replaced by the combination of cldss 23377 and eqid 2763. (Contributed by Zhi Wang, 6-Sep-2024.)
⊢ (𝑍 = (∪ 𝐽 ∖ 𝑋) → (𝑌 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑍) ↔ (𝑌 ∈ (Clsd‘𝐽) ∧ (𝑋 ∩ 𝑌) = ∅)))
21.53.10.2  Neighborhoods
Theorem	neircl 50224	Reverse closure of the neighborhood operation. (This theorem depends on a function's value being empty outside of its domain, but it will make later theorems simpler to state.) (Contributed by Zhi Wang, 16-Sep-2024.)
⊢ (𝑁 ∈ ((nei‘𝐽)‘𝑆) → 𝐽 ∈ Top)
Theorem	opnneilem 50225*	Lemma factoring out common proof steps of opnneil 50229 and opnneirv 50227. (Contributed by Zhi Wang, 31-Aug-2024.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐽 (𝑆 ⊆ 𝑥 ∧ 𝜓) ↔ ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝜒)))
Theorem	opnneir 50226*	If something is true for an open neighborhood, it must be true for a neighborhood. (Contributed by Zhi Wang, 31-Aug-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐽 (𝑆 ⊆ 𝑥 ∧ 𝜓) → ∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓))
Theorem	opnneirv 50227*	A variant of opnneir 50226 with different dummy variables. (Contributed by Zhi Wang, 31-Aug-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐽 (𝑆 ⊆ 𝑥 ∧ 𝜓) → ∃𝑦 ∈ ((nei‘𝐽)‘𝑆)𝜒))
Theorem	opnneilv 50228*	The converse of opnneir 50226 with different dummy variables. Note that the second hypothesis could be generalized by adding 𝑦 ∈ 𝐽 to the antecedent. See the proof for details. Although 𝐽 ∈ Top might be redundant here (see neircl 50224), it is listed for explicitness. (Contributed by Zhi Wang, 31-Aug-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝑥) → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 → ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝜒)))
Theorem	opnneil 50229*	A variant of opnneilv 50228. (Contributed by Zhi Wang, 31-Aug-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝑥) → (𝜓 → 𝜒))    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 → ∃𝑥 ∈ 𝐽 (𝑆 ⊆ 𝑥 ∧ 𝜓)))
Theorem	opnneibid 50230*	The equivalence between neighborhood and open neighborhood. See opnneibid2 50231 for different dummy variables. (Contributed by Zhi Wang, 31-Aug-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝑥) → (𝜓 → 𝜒))    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 ↔ ∃𝑥 ∈ 𝐽 (𝑆 ⊆ 𝑥 ∧ 𝜓)))
Theorem	opnneibid2 50231*	The equivalence between neighborhood and open neighborhood. A variant of opnneibid 50230 with two dummy variables. (Contributed by Zhi Wang, 31-Aug-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝑥) → (𝜓 → 𝜒))    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 ↔ ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝜒)))
21.53.10.3  Subspace topologies
Theorem	restcls2lem 50232	A closed set in a subspace topology is a subset of the subspace. (Contributed by Zhi Wang, 2-Sep-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝑋 = ∪ 𝐽)    &   ⊢ (𝜑 → 𝑌 ⊆ 𝑋)    &   ⊢ (𝜑 → 𝐾 = (𝐽 ↾t 𝑌))    &   ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐾))    ⇒   ⊢ (𝜑 → 𝑆 ⊆ 𝑌)
Theorem	restcls2 50233	A closed set in a subspace topology is the closure in the original topology intersecting with the subspace. (Contributed by Zhi Wang, 2-Sep-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝑋 = ∪ 𝐽)    &   ⊢ (𝜑 → 𝑌 ⊆ 𝑋)    &   ⊢ (𝜑 → 𝐾 = (𝐽 ↾t 𝑌))    &   ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐾))    ⇒   ⊢ (𝜑 → 𝑆 = (((cls‘𝐽)‘𝑆) ∩ 𝑌))
Theorem	restclsseplem 50234	Lemma for restclssep 50235. (Contributed by Zhi Wang, 2-Sep-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝑋 = ∪ 𝐽)    &   ⊢ (𝜑 → 𝑌 ⊆ 𝑋)    &   ⊢ (𝜑 → 𝐾 = (𝐽 ↾t 𝑌))    &   ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐾))    &   ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅)    &   ⊢ (𝜑 → 𝑇 ⊆ 𝑌)    ⇒   ⊢ (𝜑 → (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)
Theorem	restclssep 50235	Two disjoint closed sets in a subspace topology are separated in the original topology. (Contributed by Zhi Wang, 2-Sep-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝑋 = ∪ 𝐽)    &   ⊢ (𝜑 → 𝑌 ⊆ 𝑋)    &   ⊢ (𝜑 → 𝐾 = (𝐽 ↾t 𝑌))    &   ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐾))    &   ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅)    &   ⊢ (𝜑 → 𝑇 ∈ (Clsd‘𝐾))    ⇒   ⊢ (𝜑 → ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅))
21.53.10.4  Limits and continuity in topological spaces
Theorem	cnneiima 50236	Given a continuous function, the preimage of a neighborhood is a neighborhood. To be precise, the preimage of a neighborhood of a subset 𝑇 of the codomain of a continuous function is a neighborhood of any subset of the preimage of 𝑇. (Contributed by Zhi Wang, 9-Sep-2024.)
⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝑁 ∈ ((nei‘𝐾)‘𝑇))    &   ⊢ (𝜑 → 𝑆 ⊆ (◡𝐹 “ 𝑇))    ⇒   ⊢ (𝜑 → (◡𝐹 “ 𝑁) ∈ ((nei‘𝐽)‘𝑆))
21.53.10.5  Topological definitions using the reals
Theorem	iooii 50237	Open intervals are open sets of II. (Contributed by Zhi Wang, 9-Sep-2024.)
⊢ ((0 ≤ 𝐴 ∧ 𝐵 ≤ 1) → (𝐴(,)𝐵) ∈ II)
Theorem	icccldii 50238	Closed intervals are closed sets of II. Note that iccss 13572, iccordt 23562, and ordtresticc 23571 are proved from ixxss12 13523, ordtcld3 23547, and ordtrest2 23552, respectively. An alternate proof uses restcldi 23521, dfii2 25233, and icccld 25115. (Contributed by Zhi Wang, 8-Sep-2024.)
⊢ ((0 ≤ 𝐴 ∧ 𝐵 ≤ 1) → (𝐴[,]𝐵) ∈ (Clsd‘II))
Theorem	i0oii 50239	(0[,)𝐴) is open in II. (Contributed by Zhi Wang, 9-Sep-2024.)
⊢ (𝐴 ≤ 1 → (0[,)𝐴) ∈ II)
Theorem	io1ii 50240	(𝐴(,]1) is open in II. (Contributed by Zhi Wang, 9-Sep-2024.)
⊢ (0 ≤ 𝐴 → (𝐴(,]1) ∈ II)
21.53.10.6  Separated sets
Theorem	sepnsepolem1 50241*	Lemma for sepnsepo 50243. (Contributed by Zhi Wang, 1-Sep-2024.)
⊢ (∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ∃𝑥 ∈ 𝐽 (𝜑 ∧ ∃𝑦 ∈ 𝐽 (𝜓 ∧ 𝜒)))
Theorem	sepnsepolem2 50242*	Open neighborhood and neighborhood is equivalent regarding disjointness. Lemma for sepnsepo 50243. Proof could be shortened by 1 step using ssdisjdr 50124. (Contributed by Zhi Wang, 1-Sep-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    ⇒   ⊢ (𝜑 → (∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥 ∩ 𝑦) = ∅ ↔ ∃𝑦 ∈ 𝐽 (𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)))
Theorem	sepnsepo 50243*	Open neighborhood and neighborhood is equivalent regarding disjointness for both sides. Namely, separatedness by open neighborhoods is equivalent to separatedness by neighborhoods. (Contributed by Zhi Wang, 1-Sep-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝐶)∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥 ∩ 𝑦) = ∅ ↔ ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ 𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)))
Theorem	sepdisj 50244	Separated sets are disjoint. Note that in general separatedness also requires 𝑇 ⊆ ∪ 𝐽 and (𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ as well but they are unnecessary here. (Contributed by Zhi Wang, 7-Sep-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽)    &   ⊢ (𝜑 → (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)    ⇒   ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅)
Theorem	seposep 50245*	If two sets are separated by (open) neighborhoods, then they are separated subsets of the underlying set. Note that separatedness by open neighborhoods is equivalent to separatedness by neighborhoods. See sepnsepo 50243. The relationship between separatedness and closure is also seen in isnrm 23683, isnrm2 23706, isnrm3 23707. (Contributed by Zhi Wang, 7-Sep-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))    ⇒   ⊢ (𝜑 → ((𝑆 ⊆ ∪ 𝐽 ∧ 𝑇 ⊆ ∪ 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)))
Theorem	sepcsepo 50246*	If two sets are separated by closed neighborhoods, then they are separated by (open) neighborhoods. See sepnsepo 50243 for the equivalence between separatedness by open neighborhoods and separatedness by neighborhoods. Although 𝐽 ∈ Top might be redundant here, it is listed for explicitness. 𝐽 ∈ Top can be obtained from neircl 50224, adantr 486, and rexlimiva 3159. (Contributed by Zhi Wang, 8-Sep-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → ∃𝑛 ∈ ((nei‘𝐽)‘𝑆)∃𝑚 ∈ ((nei‘𝐽)‘𝑇)(𝑛 ∈ (Clsd‘𝐽) ∧ 𝑚 ∈ (Clsd‘𝐽) ∧ (𝑛 ∩ 𝑚) = ∅))    ⇒   ⊢ (𝜑 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))
Theorem	sepfsepc 50247*	If two sets are separated by a continuous function, then they are separated by closed neighborhoods. (Contributed by Zhi Wang, 9-Sep-2024.)
⊢ (𝜑 → ∃𝑓 ∈ (𝐽 Cn II)(𝑆 ⊆ (◡𝑓 “ {0}) ∧ 𝑇 ⊆ (◡𝑓 “ {1})))    ⇒   ⊢ (𝜑 → ∃𝑛 ∈ ((nei‘𝐽)‘𝑆)∃𝑚 ∈ ((nei‘𝐽)‘𝑇)(𝑛 ∈ (Clsd‘𝐽) ∧ 𝑚 ∈ (Clsd‘𝐽) ∧ (𝑛 ∩ 𝑚) = ∅))
Theorem	seppsepf 50248	If two sets are precisely separated by a continuous function, then they are separated by the continuous function. (Contributed by Zhi Wang, 9-Sep-2024.)
⊢ (𝜑 → ∃𝑓 ∈ (𝐽 Cn II)(𝑆 = (◡𝑓 “ {0}) ∧ 𝑇 = (◡𝑓 “ {1})))    ⇒   ⊢ (𝜑 → ∃𝑓 ∈ (𝐽 Cn II)(𝑆 ⊆ (◡𝑓 “ {0}) ∧ 𝑇 ⊆ (◡𝑓 “ {1})))
Theorem	seppcld 50249*	If two sets are precisely separated by a continuous function, then they are closed. An alternate proof involves II ∈ Fre. (Contributed by Zhi Wang, 9-Sep-2024.)
⊢ (𝜑 → ∃𝑓 ∈ (𝐽 Cn II)(𝑆 = (◡𝑓 “ {0}) ∧ 𝑇 = (◡𝑓 “ {1})))    ⇒   ⊢ (𝜑 → (𝑆 ∈ (Clsd‘𝐽) ∧ 𝑇 ∈ (Clsd‘𝐽)))
21.53.10.7  Separated spaces: T0, T1, T2 (Hausdorff) ...
Theorem	isnrm4 50250*	A topological space is normal iff any two disjoint closed sets are separated by neighborhoods. (Contributed by Zhi Wang, 1-Sep-2024.)
⊢ (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ ((nei‘𝐽)‘𝑐)∃𝑦 ∈ ((nei‘𝐽)‘𝑑)(𝑥 ∩ 𝑦) = ∅)))
Theorem	dfnrm2 50251*	A topological space is normal if any disjoint closed sets can be separated by open neighborhoods. An alternate definition of df-nrm 23665. (Contributed by Zhi Wang, 30-Aug-2024.)
⊢ Nrm = {𝑗 ∈ Top ∣ ∀𝑐 ∈ (Clsd‘𝑗)∀𝑑 ∈ (Clsd‘𝑗)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝑗 ∃𝑦 ∈ 𝑗 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))}
Theorem	dfnrm3 50252*	A topological space is normal if any disjoint closed sets can be separated by neighborhoods. An alternate definition of df-nrm 23665. (Contributed by Zhi Wang, 2-Sep-2024.)
⊢ Nrm = {𝑗 ∈ Top ∣ ∀𝑐 ∈ (Clsd‘𝑗)∀𝑑 ∈ (Clsd‘𝑗)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ ((nei‘𝑗)‘𝑐)∃𝑦 ∈ ((nei‘𝑗)‘𝑑)(𝑥 ∩ 𝑦) = ∅)}
Theorem	iscnrm3lem1 50253*	Lemma for iscnrm3 50271. Subspace topology is a topology. (Contributed by Zhi Wang, 3-Sep-2024.)
⊢ (𝐽 ∈ Top → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ((𝐽 ↾t 𝑥) ∈ Top ∧ 𝜑)))
Theorem	iscnrm3lem2 50254*	Lemma for iscnrm3 50271 proving a biconditional on restricted universal quantifications. (Contributed by Zhi Wang, 3-Sep-2024.)
⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓 → ((𝑤 ∈ 𝐷 ∧ 𝑣 ∈ 𝐸) → 𝜒)))    &   ⊢ (𝜑 → (∀𝑤 ∈ 𝐷 ∀𝑣 ∈ 𝐸 𝜒 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → 𝜓)))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓 ↔ ∀𝑤 ∈ 𝐷 ∀𝑣 ∈ 𝐸 𝜒))
Theorem	iscnrm3lem4 50255	Lemma for iscnrm3lem5 50256 and iscnrm3r 50267. (Contributed by Zhi Wang, 4-Sep-2024.)
⊢ (𝜂 → (𝜓 → 𝜁))    &   ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜂)    &   ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → (𝜁 → 𝜏))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → 𝜏))))
Theorem	iscnrm3lem5 50256*	Lemma for iscnrm3l 50270. (Contributed by Zhi Wang, 3-Sep-2024.)
⊢ ((𝑥 = 𝑆 ∧ 𝑦 = 𝑇) → (𝜑 ↔ 𝜓))    &   ⊢ ((𝑥 = 𝑆 ∧ 𝑦 = 𝑇) → (𝜒 ↔ 𝜃))    &   ⊢ ((𝜏 ∧ 𝜂 ∧ 𝜁) → (𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊))    &   ⊢ ((𝜏 ∧ 𝜂 ∧ 𝜁) → ((𝜓 → 𝜃) → 𝜎))    ⇒   ⊢ (𝜏 → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑊 (𝜑 → 𝜒) → (𝜂 → (𝜁 → 𝜎))))
Theorem	iscnrm3lem6 50257*	Lemma for iscnrm3lem7 50258. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑊) ∧ 𝜓) → 𝜒)    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑊 𝜓 → 𝜒))
Theorem	iscnrm3lem7 50258*	Lemma for iscnrm3rlem8 50266 and iscnrm3llem2 50269 involving restricted existential quantifications. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ (𝑧 = 𝑍 → (𝜒 ↔ 𝜃))    &   ⊢ (𝑤 = 𝑊 → (𝜃 ↔ 𝜏))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓) → (𝑍 ∈ 𝐶 ∧ 𝑊 ∈ 𝐷 ∧ 𝜏))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 → ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐷 𝜒))
Theorem	iscnrm3rlem1 50259	Lemma for iscnrm3rlem2 50260. The hypothesis could be generalized to (𝜑 → (𝑆 ∖ 𝑇) ⊆ 𝑋). (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ (𝜑 → 𝑆 ⊆ 𝑋)    ⇒   ⊢ (𝜑 → (𝑆 ∖ 𝑇) = (𝑆 ∩ (𝑋 ∖ (𝑆 ∩ 𝑇))))
Theorem	iscnrm3rlem2 50260	Lemma for iscnrm3rlem3 50261. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽)    ⇒   ⊢ (𝜑 → (((cls‘𝐽)‘𝑆) ∖ 𝑇) ∈ (Clsd‘(𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ 𝑇)))))
Theorem	iscnrm3rlem3 50261	Lemma for iscnrm3r 50267. The designed subspace is a subset of the original set; the two sets are closed sets in the subspace. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ ((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽)) → ((∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ∈ 𝒫 ∪ 𝐽 ∧ (((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ∈ (Clsd‘(𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ∈ (Clsd‘(𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))))
Theorem	iscnrm3rlem4 50262	Lemma for iscnrm3rlem8 50266. Given two disjoint subsets 𝑆 and 𝑇 of the underlying set of a topology 𝐽, if 𝑁 is a superset of (((cls‘𝐽)‘𝑆) ∖ 𝑇), then it is a superset of 𝑆. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽)    &   ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅)    &   ⊢ (𝜑 → (((cls‘𝐽)‘𝑆) ∖ 𝑇) ⊆ 𝑁)    ⇒   ⊢ (𝜑 → 𝑆 ⊆ 𝑁)
Theorem	iscnrm3rlem5 50263	Lemma for iscnrm3rlem6 50264. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽)    &   ⊢ (𝜑 → 𝑇 ⊆ ∪ 𝐽)    ⇒   ⊢ (𝜑 → (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ∈ 𝐽)
Theorem	iscnrm3rlem6 50264	Lemma for iscnrm3rlem7 50265. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽)    &   ⊢ (𝜑 → 𝑇 ⊆ ∪ 𝐽)    &   ⊢ (𝜑 → 𝑂 ⊆ (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))    ⇒   ⊢ (𝜑 → (𝑂 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) ↔ 𝑂 ∈ 𝐽))
Theorem	iscnrm3rlem7 50265	Lemma for iscnrm3rlem8 50266. Open neighborhoods in the subspace topology are open neighborhoods in the original topology given that the subspace is an open set in the original topology. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽)    &   ⊢ (𝜑 → 𝑇 ⊆ ∪ 𝐽)    &   ⊢ (𝜑 → 𝑂 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))    ⇒   ⊢ (𝜑 → 𝑂 ∈ 𝐽)
Theorem	iscnrm3rlem8 50266*	Lemma for iscnrm3r 50267. Disjoint open neighborhoods in the subspace topology are disjoint open neighborhoods in the original topology given that the subspace is an open set in the original topology. Therefore, given any two sets separated in the original topology and separated by open neighborhoods in the subspace topology, they must be separated by open neighborhoods in the original topology. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ ((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)) → (∃𝑙 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))∃𝑘 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ⊆ 𝑙 ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)))
Theorem	iscnrm3r 50267*	Lemma for iscnrm3 50271. If all subspaces of a topology are normal, i.e., two disjoint closed sets can be separated by open neighborhoods, then in the original topology two separated sets can be separated by open neighborhoods. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ (𝐽 ∈ Top → (∀𝑧 ∈ 𝒫 ∪ 𝐽∀𝑐 ∈ (Clsd‘(𝐽 ↾t 𝑧))∀𝑑 ∈ (Clsd‘(𝐽 ↾t 𝑧))((𝑐 ∩ 𝑑) = ∅ → ∃𝑙 ∈ (𝐽 ↾t 𝑧)∃𝑘 ∈ (𝐽 ↾t 𝑧)(𝑐 ⊆ 𝑙 ∧ 𝑑 ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)) → ((𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽) → (((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)))))
Theorem	iscnrm3llem1 50268	Lemma for iscnrm3l 50270. Closed sets in the subspace are subsets of the underlying set of the original topology. (Contributed by Zhi Wang, 4-Sep-2024.)
⊢ ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) → (𝐶 ∈ 𝒫 ∪ 𝐽 ∧ 𝐷 ∈ 𝒫 ∪ 𝐽))
Theorem	iscnrm3llem2 50269*	Lemma for iscnrm3l 50270. If there exist disjoint open neighborhoods in the original topology for two disjoint closed sets in a subspace, then they can be separated by open neighborhoods in the subspace topology. (Could shorten proof with ssin0 46277.) (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) → (∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝐶 ⊆ 𝑛 ∧ 𝐷 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅) → ∃𝑙 ∈ (𝐽 ↾t 𝑍)∃𝑘 ∈ (𝐽 ↾t 𝑍)(𝐶 ⊆ 𝑙 ∧ 𝐷 ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)))
Theorem	iscnrm3l 50270*	Lemma for iscnrm3 50271. Given a topology 𝐽, if two separated sets can be separated by open neighborhoods, then all subspaces of the topology 𝐽 are normal, i.e., two disjoint closed sets can be separated by open neighborhoods. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ (𝐽 ∈ Top → (∀𝑠 ∈ 𝒫 ∪ 𝐽∀𝑡 ∈ 𝒫 ∪ 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → ((𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) → ((𝐶 ∩ 𝐷) = ∅ → ∃𝑙 ∈ (𝐽 ↾t 𝑍)∃𝑘 ∈ (𝐽 ↾t 𝑍)(𝐶 ⊆ 𝑙 ∧ 𝐷 ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)))))
Theorem	iscnrm3 50271*	A completely normal topology is a topology in which two separated sets can be separated by open neighborhoods. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑠 ∈ 𝒫 ∪ 𝐽∀𝑡 ∈ 𝒫 ∪ 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))))
Theorem	iscnrm3v 50272*	A topology is completely normal iff two separated sets can be separated by open neighborhoods. (Contributed by Zhi Wang, 10-Sep-2024.)
⊢ (𝐽 ∈ Top → (𝐽 ∈ CNrm ↔ ∀𝑠 ∈ 𝒫 ∪ 𝐽∀𝑡 ∈ 𝒫 ∪ 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))))
Theorem	iscnrm4 50273*	A completely normal topology is a topology in which two separated sets can be separated by neighborhoods. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑠 ∈ 𝒫 ∪ 𝐽∀𝑡 ∈ 𝒫 ∪ 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ ((nei‘𝐽)‘𝑠)∃𝑚 ∈ ((nei‘𝐽)‘𝑡)(𝑛 ∩ 𝑚) = ∅)))
21.53.11  Preordered sets and directed sets using extensible structures
Theorem	isprsd 50274*	Property of being a preordered set (deduction form). (Contributed by Zhi Wang, 18-Sep-2024.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → ≤ = (le‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐾 ∈ Proset ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧))))
21.53.12  Posets and lattices using extensible structures
21.53.12.1  Posets
Theorem	lubeldm2 50275*	Member of the domain of the least upper bound function of a poset. (Contributed by Zhi Wang, 26-Sep-2024.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝑈 = (lub‘𝐾)    &   ⊢ (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))    &   ⊢ (𝜑 → 𝐾 ∈ Poset)    ⇒   ⊢ (𝜑 → (𝑆 ∈ dom 𝑈 ↔ (𝑆 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝜓)))
Theorem	glbeldm2 50276*	Member of the domain of the greatest lower bound function of a poset. (Contributed by Zhi Wang, 26-Sep-2024.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐺 = (glb‘𝐾)    &   ⊢ (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))    &   ⊢ (𝜑 → 𝐾 ∈ Poset)    ⇒   ⊢ (𝜑 → (𝑆 ∈ dom 𝐺 ↔ (𝑆 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝜓)))
Theorem	lubeldm2d 50277*	Member of the domain of the least upper bound function of a poset. (Contributed by Zhi Wang, 28-Sep-2024.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → ≤ = (le‘𝐾))    &   ⊢ (𝜑 → 𝑈 = (lub‘𝐾))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))))    &   ⊢ (𝜑 → 𝐾 ∈ Poset)    ⇒   ⊢ (𝜑 → (𝑆 ∈ dom 𝑈 ↔ (𝑆 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝜓)))
Theorem	glbeldm2d 50278*	Member of the domain of the greatest lower bound function of a poset. (Contributed by Zhi Wang, 29-Sep-2024.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → ≤ = (le‘𝐾))    &   ⊢ (𝜑 → 𝐺 = (glb‘𝐾))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))))    &   ⊢ (𝜑 → 𝐾 ∈ Poset)    ⇒   ⊢ (𝜑 → (𝑆 ∈ dom 𝐺 ↔ (𝑆 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝜓)))
Theorem	lubsscl 50279	If a subset of 𝑆 contains the LUB of 𝑆, then the two sets have the same LUB. (Contributed by Zhi Wang, 26-Sep-2024.)
⊢ (𝜑 → 𝐾 ∈ Poset)    &   ⊢ (𝜑 → 𝑇 ⊆ 𝑆)    &   ⊢ 𝑈 = (lub‘𝐾)    &   ⊢ (𝜑 → 𝑆 ∈ dom 𝑈)    &   ⊢ (𝜑 → (𝑈‘𝑆) ∈ 𝑇)    ⇒   ⊢ (𝜑 → (𝑇 ∈ dom 𝑈 ∧ (𝑈‘𝑇) = (𝑈‘𝑆)))
Theorem	glbsscl 50280	If a subset of 𝑆 contains the GLB of 𝑆, then the two sets have the same GLB. (Contributed by Zhi Wang, 26-Sep-2024.)
⊢ (𝜑 → 𝐾 ∈ Poset)    &   ⊢ (𝜑 → 𝑇 ⊆ 𝑆)    &   ⊢ 𝐺 = (glb‘𝐾)    &   ⊢ (𝜑 → 𝑆 ∈ dom 𝐺)    &   ⊢ (𝜑 → (𝐺‘𝑆) ∈ 𝑇)    ⇒   ⊢ (𝜑 → (𝑇 ∈ dom 𝐺 ∧ (𝐺‘𝑇) = (𝐺‘𝑆)))
Theorem	lubprlem 50281	Lemma for lubprdm 50282 and lubpr 50283. (Contributed by Zhi Wang, 26-Sep-2024.)
⊢ (𝜑 → 𝐾 ∈ Poset)    &   ⊢ 𝐵 = (Base‘𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ (𝜑 → 𝑋 ≤ 𝑌)    &   ⊢ (𝜑 → 𝑆 = {𝑋, 𝑌})    &   ⊢ 𝑈 = (lub‘𝐾)    ⇒   ⊢ (𝜑 → (𝑆 ∈ dom 𝑈 ∧ (𝑈‘𝑆) = 𝑌))
Theorem	lubprdm 50282	The set of two comparable elements in a poset has LUB. (Contributed by Zhi Wang, 26-Sep-2024.)
⊢ (𝜑 → 𝐾 ∈ Poset)    &   ⊢ 𝐵 = (Base‘𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ (𝜑 → 𝑋 ≤ 𝑌)    &   ⊢ (𝜑 → 𝑆 = {𝑋, 𝑌})    &   ⊢ 𝑈 = (lub‘𝐾)    ⇒   ⊢ (𝜑 → 𝑆 ∈ dom 𝑈)
Theorem	lubpr 50283	The LUB of the set of two comparable elements in a poset is the greater one of the two. (Contributed by Zhi Wang, 26-Sep-2024.)
⊢ (𝜑 → 𝐾 ∈ Poset)    &   ⊢ 𝐵 = (Base‘𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ (𝜑 → 𝑋 ≤ 𝑌)    &   ⊢ (𝜑 → 𝑆 = {𝑋, 𝑌})    &   ⊢ 𝑈 = (lub‘𝐾)    ⇒   ⊢ (𝜑 → (𝑈‘𝑆) = 𝑌)
Theorem	glbprlem 50284	Lemma for glbprdm 50285 and glbpr 50286. (Contributed by Zhi Wang, 26-Sep-2024.)
⊢ (𝜑 → 𝐾 ∈ Poset)    &   ⊢ 𝐵 = (Base‘𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ (𝜑 → 𝑋 ≤ 𝑌)    &   ⊢ (𝜑 → 𝑆 = {𝑋, 𝑌})    &   ⊢ 𝐺 = (glb‘𝐾)    ⇒   ⊢ (𝜑 → (𝑆 ∈ dom 𝐺 ∧ (𝐺‘𝑆) = 𝑋))
Theorem	glbprdm 50285	The set of two comparable elements in a poset has GLB. (Contributed by Zhi Wang, 26-Sep-2024.)
⊢ (𝜑 → 𝐾 ∈ Poset)    &   ⊢ 𝐵 = (Base‘𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ (𝜑 → 𝑋 ≤ 𝑌)    &   ⊢ (𝜑 → 𝑆 = {𝑋, 𝑌})    &   ⊢ 𝐺 = (glb‘𝐾)    ⇒   ⊢ (𝜑 → 𝑆 ∈ dom 𝐺)
Theorem	glbpr 50286	The GLB of the set of two comparable elements in a poset is the less one of the two. (Contributed by Zhi Wang, 26-Sep-2024.)
⊢ (𝜑 → 𝐾 ∈ Poset)    &   ⊢ 𝐵 = (Base‘𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ (𝜑 → 𝑋 ≤ 𝑌)    &   ⊢ (𝜑 → 𝑆 = {𝑋, 𝑌})    &   ⊢ 𝐺 = (glb‘𝐾)    ⇒   ⊢ (𝜑 → (𝐺‘𝑆) = 𝑋)
Theorem	joindm2 50287*	The join of any two elements always exists iff all unordered pairs have LUB. (Contributed by Zhi Wang, 25-Sep-2024.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑉)    &   ⊢ 𝑈 = (lub‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    ⇒   ⊢ (𝜑 → (dom ∨ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ dom 𝑈))
Theorem	joindm3 50288*	The join of any two elements always exists iff all unordered pairs have LUB (expanded version). (Contributed by Zhi Wang, 25-Sep-2024.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑉)    &   ⊢ 𝑈 = (lub‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    ⇒   ⊢ (𝜑 → (dom ∨ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃!𝑧 ∈ 𝐵 ((𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧) ∧ ∀𝑤 ∈ 𝐵 ((𝑥 ≤ 𝑤 ∧ 𝑦 ≤ 𝑤) → 𝑧 ≤ 𝑤))))
Theorem	meetdm2 50289*	The meet of any two elements always exists iff all unordered pairs have GLB. (Contributed by Zhi Wang, 25-Sep-2024.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑉)    &   ⊢ 𝐺 = (glb‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    ⇒   ⊢ (𝜑 → (dom ∧ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ dom 𝐺))
Theorem	meetdm3 50290*	The meet of any two elements always exists iff all unordered pairs have GLB (expanded version). (Contributed by Zhi Wang, 25-Sep-2024.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑉)    &   ⊢ 𝐺 = (glb‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    ⇒   ⊢ (𝜑 → (dom ∧ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃!𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑥 ∧ 𝑧 ≤ 𝑦) ∧ ∀𝑤 ∈ 𝐵 ((𝑤 ≤ 𝑥 ∧ 𝑤 ≤ 𝑦) → 𝑤 ≤ 𝑧))))
Theorem	posjidm 50291	Poset join is idempotent. latjidm 18665 could be shortened by this. (Contributed by Zhi Wang, 27-Sep-2024.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → (𝑋 ∨ 𝑋) = 𝑋)
Theorem	posmidm 50292	Poset meet is idempotent. latmidm 18677 could be shortened by this. (Contributed by Zhi Wang, 27-Sep-2024.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ 𝑋) = 𝑋)
Theorem	resiposbas 50293	Construct a poset (resipos 50294) for any base set. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ 𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(le‘ndx), ( I ↾ 𝐵)⟩}    ⇒   ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐾))
Theorem	resipos 50294	A set equipped with an order where no distinct elements are comparable is a poset. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ 𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(le‘ndx), ( I ↾ 𝐵)⟩}    ⇒   ⊢ (𝐵 ∈ 𝑉 → 𝐾 ∈ Poset)
Theorem	exbaspos 50295*	There exists a poset for any base set. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ (𝐵 ∈ 𝑉 → ∃𝑘 ∈ Poset 𝐵 = (Base‘𝑘))
Theorem	exbasprs 50296*	There exists a preordered set for any base set. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ (𝐵 ∈ 𝑉 → ∃𝑘 ∈ Proset 𝐵 = (Base‘𝑘))
Theorem	basresposfo 50297	The base function restricted to the class of posets maps the class of posets onto the universal class. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ (Base ↾ Poset):Poset⟶ontoV
Theorem	basresprsfo 50298	The base function restricted to the class of preordered sets maps the class of preordered sets onto the universal class. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ (Base ↾ Proset ): Proset ⟶ontoV
Theorem	posnex 50299	The class of posets is a proper class. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ Poset ∉ V
Theorem	prsnex 50300	The class of preordered sets is a proper class. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ Proset ∉ V