P. 491 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 49001-49100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	opncldeqv 49001*	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 49002	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 49003	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 49002 with elssuni 4887 replaced by the combination of cldss 22944 and eqid 2731. (Contributed by Zhi Wang, 6-Sep-2024.)
⊢ (𝑍 = (∪ 𝐽 ∖ 𝑋) → (𝑌 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑍) ↔ (𝑌 ∈ (Clsd‘𝐽) ∧ (𝑋 ∩ 𝑌) = ∅)))
21.49.10.2  Neighborhoods
Theorem	neircl 49004	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 49005*	Lemma factoring out common proof steps of opnneil 49009 and opnneirv 49007. (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 49006*	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 49007*	A variant of opnneir 49006 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 49008*	The converse of opnneir 49006 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 49004), 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 49009*	A variant of opnneilv 49008. (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	opnneieqv 49010*	The equivalence between neighborhood and open neighborhood. See opnneieqvv 49011 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	opnneieqvv 49011*	The equivalence between neighborhood and open neighborhood. A variant of opnneieqv 49010 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)))
21.49.10.3  Subspace topologies
Theorem	restcls2lem 49012	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 49013	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 49014	Lemma for restclssep 49015. (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 49015	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) = ∅))
21.49.10.4  Limits and continuity in topological spaces
Theorem	cnneiima 49016	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))
21.49.10.5  Topological definitions using the reals
Theorem	iooii 49017	Open intervals are open sets of II. (Contributed by Zhi Wang, 9-Sep-2024.)
⊢ ((0 ≤ 𝐴 ∧ 𝐵 ≤ 1) → (𝐴(,)𝐵) ∈ II)
Theorem	icccldii 49018	Closed intervals are closed sets of II. Note that iccss 13314, iccordt 23129, and ordtresticc 23138 are proved from ixxss12 13265, ordtcld3 23114, and ordtrest2 23119, respectively. An alternate proof uses restcldi 23088, dfii2 24802, and icccld 24681. (Contributed by Zhi Wang, 8-Sep-2024.)
⊢ ((0 ≤ 𝐴 ∧ 𝐵 ≤ 1) → (𝐴[,]𝐵) ∈ (Clsd‘II))
Theorem	i0oii 49019	(0[,)𝐴) is open in II. (Contributed by Zhi Wang, 9-Sep-2024.)
⊢ (𝐴 ≤ 1 → (0[,)𝐴) ∈ II)
Theorem	io1ii 49020	(𝐴(,]1) is open in II. (Contributed by Zhi Wang, 9-Sep-2024.)
⊢ (0 ≤ 𝐴 → (𝐴(,]1) ∈ II)
21.49.10.6  Separated sets
Theorem	sepnsepolem1 49021*	Lemma for sepnsepo 49023. (Contributed by Zhi Wang, 1-Sep-2024.)
⊢ (∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ∃𝑥 ∈ 𝐽 (𝜑 ∧ ∃𝑦 ∈ 𝐽 (𝜓 ∧ 𝜒)))
Theorem	sepnsepolem2 49022*	Open neighborhood and neighborhood is equivalent regarding disjointness. Lemma for sepnsepo 49023. Proof could be shortened by 1 step using ssdisjdr 48908. (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 49023*	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 49024	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 49025*	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 49023. The relationship between separatedness and closure is also seen in isnrm 23250, isnrm2 23273, isnrm3 23274. (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 49026*	If two sets are separated by closed neighborhoods, then they are separated by (open) neighborhoods. See sepnsepo 49023 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 49004, adantr 480, and rexlimiva 3125. (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 49027*	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 49028	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 49029*	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)))
21.49.10.7  Separated spaces: T0, T1, T2 (Hausdorff) ...
Theorem	isnrm4 49030*	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 49031*	A topological space is normal if any disjoint closed sets can be separated by open neighborhoods. An alternate definition of df-nrm 23232. (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 49032*	A topological space is normal if any disjoint closed sets can be separated by neighborhoods. An alternate definition of df-nrm 23232. (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 49033*	Lemma for iscnrm3 49051. 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 49034*	Lemma for iscnrm3 49051 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 49035	Lemma for iscnrm3lem5 49036 and iscnrm3r 49047. (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 49036*	Lemma for iscnrm3l 49050. (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 49037*	Lemma for iscnrm3lem7 49038. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑊) ∧ 𝜓) → 𝜒)    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑊 𝜓 → 𝜒))
Theorem	iscnrm3lem7 49038*	Lemma for iscnrm3rlem8 49046 and iscnrm3llem2 49049 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 49039	Lemma for iscnrm3rlem2 49040. 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 49040	Lemma for iscnrm3rlem3 49041. (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 49041	Lemma for iscnrm3r 49047. 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 49042	Lemma for iscnrm3rlem8 49046. 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 49043	Lemma for iscnrm3rlem6 49044. (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 49044	Lemma for iscnrm3rlem7 49045. (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 49045	Lemma for iscnrm3rlem8 49046. 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 49046*	Lemma for iscnrm3r 49047. 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 49047*	Lemma for iscnrm3 49051. 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 49048	Lemma for iscnrm3l 49050. 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 49049*	Lemma for iscnrm3l 49050. 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 45151.) (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 49050*	Lemma for iscnrm3 49051. 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 49051*	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 49052*	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 49053*	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) = ∅)))
21.49.11  Preordered sets and directed sets using extensible structures
Theorem	isprsd 49054*	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))))
21.49.12  Posets and lattices using extensible structures
21.49.12.1  Posets
Theorem	lubeldm2 49055*	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 49056*	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 49057*	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 49058*	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 49059	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 49060	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 49061	Lemma for lubprdm 49062 and lubpr 49063. (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 49062	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 49063	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 49064	Lemma for glbprdm 49065 and glbpr 49066. (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 49065	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 49066	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 49067*	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 49068*	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 49069*	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 49070*	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 49071	Poset join is idempotent. latjidm 18368 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 49072	Poset meet is idempotent. latmidm 18380 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 49073	Construct a poset (resipos 49074) 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 49074	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 49075*	There exists a poset for any base set. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ (𝐵 ∈ 𝑉 → ∃𝑘 ∈ Poset 𝐵 = (Base‘𝑘))
Theorem	exbasprs 49076*	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 49077	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–onto→V
Theorem	basresprsfo 49078	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 –onto→V
Theorem	posnex 49079	The class of posets is a proper class. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ Poset ∉ V
Theorem	prsnex 49080	The class of preordered sets is a proper class. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ Proset ∉ V
21.49.12.2  Lattices
Theorem	toslat 49081	A toset is a lattice. (Contributed by Zhi Wang, 26-Sep-2024.)
⊢ (𝐾 ∈ Toset → 𝐾 ∈ Lat)
Theorem	isclatd 49082*	The predicate "is a complete lattice" (deduction form). (Contributed by Zhi Wang, 29-Sep-2024.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝑈 = (lub‘𝐾))    &   ⊢ (𝜑 → 𝐺 = (glb‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Poset)    &   ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐵) → 𝑠 ∈ dom 𝑈)    &   ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐵) → 𝑠 ∈ dom 𝐺)    ⇒   ⊢ (𝜑 → 𝐾 ∈ CLat)
21.49.12.3  Subset order structures
Theorem	intubeu 49083*	Existential uniqueness of the least upper bound. (Contributed by Zhi Wang, 28-Sep-2024.)
⊢ (𝐶 ∈ 𝐵 → ((𝐴 ⊆ 𝐶 ∧ ∀𝑦 ∈ 𝐵 (𝐴 ⊆ 𝑦 → 𝐶 ⊆ 𝑦)) ↔ 𝐶 = ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥}))
Theorem	unilbeu 49084*	Existential uniqueness of the greatest lower bound. (Contributed by Zhi Wang, 29-Sep-2024.)
⊢ (𝐶 ∈ 𝐵 → ((𝐶 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶)) ↔ 𝐶 = ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴}))
Theorem	ipolublem 49085*	Lemma for ipolubdm 49086 and ipolub 49087. (Contributed by Zhi Wang, 28-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐹)    &   ⊢ ≤ = (le‘𝐼)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐹) → ((∪ 𝑆 ⊆ 𝑋 ∧ ∀𝑧 ∈ 𝐹 (∪ 𝑆 ⊆ 𝑧 → 𝑋 ⊆ 𝑧)) ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑋 ∧ ∀𝑧 ∈ 𝐹 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑋 ≤ 𝑧))))
Theorem	ipolubdm 49086*	The domain of the LUB of the inclusion poset. (Contributed by Zhi Wang, 28-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐹)    &   ⊢ (𝜑 → 𝑈 = (lub‘𝐼))    &   ⊢ (𝜑 → 𝑇 = ∩ {𝑥 ∈ 𝐹 ∣ ∪ 𝑆 ⊆ 𝑥})    ⇒   ⊢ (𝜑 → (𝑆 ∈ dom 𝑈 ↔ 𝑇 ∈ 𝐹))
Theorem	ipolub 49087*	The LUB of the inclusion poset. (hypotheses "ipolub.s" and "ipolub.t" could be eliminated with 𝑆 ∈ dom 𝑈.) Could be significantly shortened if poslubdg 18318 is in quantified form. mrelatlub 18468 could potentially be shortened using this. See mrelatlubALT 49094. (Contributed by Zhi Wang, 28-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐹)    &   ⊢ (𝜑 → 𝑈 = (lub‘𝐼))    &   ⊢ (𝜑 → 𝑇 = ∩ {𝑥 ∈ 𝐹 ∣ ∪ 𝑆 ⊆ 𝑥})    &   ⊢ (𝜑 → 𝑇 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (𝑈‘𝑆) = 𝑇)
Theorem	ipoglblem 49088*	Lemma for ipoglbdm 49089 and ipoglb 49090. (Contributed by Zhi Wang, 29-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐹)    &   ⊢ ≤ = (le‘𝐼)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐹) → ((𝑋 ⊆ ∩ 𝑆 ∧ ∀𝑧 ∈ 𝐹 (𝑧 ⊆ ∩ 𝑆 → 𝑧 ⊆ 𝑋)) ↔ (∀𝑦 ∈ 𝑆 𝑋 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐹 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑋))))
Theorem	ipoglbdm 49089*	The domain of the GLB of the inclusion poset. (Contributed by Zhi Wang, 29-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐹)    &   ⊢ (𝜑 → 𝐺 = (glb‘𝐼))    &   ⊢ (𝜑 → 𝑇 = ∪ {𝑥 ∈ 𝐹 ∣ 𝑥 ⊆ ∩ 𝑆})    ⇒   ⊢ (𝜑 → (𝑆 ∈ dom 𝐺 ↔ 𝑇 ∈ 𝐹))
Theorem	ipoglb 49090*	The GLB of the inclusion poset. (hypotheses "ipolub.s" and "ipoglb.t" could be eliminated with 𝑆 ∈ dom 𝐺.) Could be significantly shortened if posglbdg 18319 is in quantified form. mrelatglb 18466 could potentially be shortened using this. See mrelatglbALT 49095. (Contributed by Zhi Wang, 29-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐹)    &   ⊢ (𝜑 → 𝐺 = (glb‘𝐼))    &   ⊢ (𝜑 → 𝑇 = ∪ {𝑥 ∈ 𝐹 ∣ 𝑥 ⊆ ∩ 𝑆})    &   ⊢ (𝜑 → 𝑇 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (𝐺‘𝑆) = 𝑇)
Theorem	ipolub0 49091	The LUB of the empty set is the intersection of the base. (Contributed by Zhi Wang, 30-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐹)    &   ⊢ (𝜑 → 𝑈 = (lub‘𝐼))    &   ⊢ (𝜑 → ∩ 𝐹 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑈‘∅) = ∩ 𝐹)
Theorem	ipolub00 49092	The LUB of the empty set is the empty set if it is contained. (Contributed by Zhi Wang, 30-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐹)    &   ⊢ (𝜑 → 𝑈 = (lub‘𝐼))    &   ⊢ (𝜑 → ∅ ∈ 𝐹)    ⇒   ⊢ (𝜑 → (𝑈‘∅) = ∅)
Theorem	ipoglb0 49093	The GLB of the empty set is the union of the base. (Contributed by Zhi Wang, 30-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐹)    &   ⊢ (𝜑 → 𝐺 = (glb‘𝐼))    &   ⊢ (𝜑 → ∪ 𝐹 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (𝐺‘∅) = ∪ 𝐹)
Theorem	mrelatlubALT 49094	Least upper bounds in a Moore space are realized by the closure of the union. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Proof shortened by Zhi Wang, 29-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐼 = (toInc‘𝐶)    &   ⊢ 𝐹 = (mrCls‘𝐶)    &   ⊢ 𝐿 = (lub‘𝐼)    ⇒   ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → (𝐿‘𝑈) = (𝐹‘∪ 𝑈))
Theorem	mrelatglbALT 49095	Greatest lower bounds in a Moore space are realized by intersections. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Proof shortened by Zhi Wang, 29-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐼 = (toInc‘𝐶)    &   ⊢ 𝐺 = (glb‘𝐼)    ⇒   ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶 ∧ 𝑈 ≠ ∅) → (𝐺‘𝑈) = ∩ 𝑈)
Theorem	mreclat 49096	A Moore space is a complete lattice under inclusion. (Contributed by Zhi Wang, 30-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐶)    ⇒   ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐼 ∈ CLat)
Theorem	topclat 49097	A topology is a complete lattice under inclusion. (Contributed by Zhi Wang, 30-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐽)    ⇒   ⊢ (𝐽 ∈ Top → 𝐼 ∈ CLat)
Theorem	toplatglb0 49098	The empty intersection in a topology is realized by the base set. (Contributed by Zhi Wang, 30-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐽)    &   ⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ 𝐺 = (glb‘𝐼)    ⇒   ⊢ (𝜑 → (𝐺‘∅) = ∪ 𝐽)
Theorem	toplatlub 49099	Least upper bounds in a topology are realized by unions. (Contributed by Zhi Wang, 30-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐽)    &   ⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐽)    &   ⊢ 𝑈 = (lub‘𝐼)    ⇒   ⊢ (𝜑 → (𝑈‘𝑆) = ∪ 𝑆)
Theorem	toplatglb 49100	Greatest lower bounds in a topology are realized by the interior of the intersection. (Contributed by Zhi Wang, 30-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐽)    &   ⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐽)    &   ⊢ 𝐺 = (glb‘𝐼)    &   ⊢ (𝜑 → 𝑆 ≠ ∅)    ⇒   ⊢ (𝜑 → (𝐺‘𝑆) = ((int‘𝐽)‘∩ 𝑆))