Theorem conventions-labels 29643

Description:

The following gives conventions used in the Metamath Proof Explorer (MPE, set.mm) regarding labels. For other conventions, see conventions 29642 and links therein.

Every statement has a unique identifying label, which serves the same purpose as an equation number in a book. We use various label naming conventions to provide easy-to-remember hints about their contents. Labels are not a 1-to-1 mapping, because that would create long names that would be difficult to remember and tedious to type. Instead, label names are relatively short while suggesting their purpose. Names are occasionally changed to make them more consistent or as we find better ways to name them. Here are a few of the label naming conventions:

• Axioms, definitions, and wff syntax. As noted earlier, axioms are named "ax-NAME", proofs of proven axioms are named "axNAME", and definitions are named "df-NAME". Wff syntax declarations have labels beginning with "w" followed by short fragment suggesting its purpose.
• Hypotheses. Hypotheses have the name of the final axiom or theorem, followed by ".", followed by a unique id (these ids are usually consecutive integers starting with 1, e.g., for rgen 3063"rgen.1 $e |- ( x e. A -> ph ) $." or letters corresponding to the (main) class variable used in the hypothesis, e.g., for mdet0 22099: "mdet0.d $e |- D = ( N maDet R ) $.").
• Common names. If a theorem has a well-known name, that name (or a short version of it) is sometimes used directly. Examples include barbara 2658 and stirling 44791.
• Principia Mathematica. Proofs of theorems from Principia Mathematica often use a special naming convention: "pm" followed by its identifier. For example, Theorem *2.27 of [WhiteheadRussell] p. 104 is named pm2.27 42.
• 19.x series of theorems. Similar to the conventions for the theorems from Principia Mathematica, theorems from Section 19 of [Margaris] p. 90 often use a special naming convention: "19." resp. "r19." (for corresponding restricted quantifier versions) followed by its identifier. For example, Theorem 38 from Section 19 of [Margaris] p. 90 is labeled 19.38 1841, and the restricted quantifier version of Theorem 21 from Section 19 of [Margaris] p. 90 is labeled r19.21 3251.
• Characters to be used for labels. Although the specification of Metamath allows for dots/periods "." in any label, it is usually used only in labels for hypotheses (see above). Exceptions are the labels of theorems from Principia Mathematica and the 19.x series of theorems from Section 19 of [Margaris] p. 90 (see above) and 0.999... 15823. Furthermore, the underscore "_" should not be used. Finally, only lower case characters should be used (except the special suffixes OLD, ALT, and ALTV mentioned in bullet point "Suffixes"), at least in main set.mm (exceptions are tolerated in mathboxes).
• Syntax label fragments. Most theorems are named using a concatenation of syntax label fragments (omitting variables) that represent the important part of the theorem's main conclusion. Almost every syntactic construct has a definition labeled "df-NAME", and normally NAME is the syntax label fragment. For example, the class difference construct (𝐴 ∖ 𝐵) is defined in df-dif 3950, and thus its syntax label fragment is "dif". Similarly, the subclass relation 𝐴 ⊆ 𝐵 has syntax label fragment "ss" because it is defined in df-ss 3964. Most theorem names follow from these fragments, for example, the theorem proving (𝐴 ∖ 𝐵) ⊆ 𝐴 involves a class difference ("dif") of a subset ("ss"), and thus is labeled difss 4130. There are many other syntax label fragments, e.g., singleton construct {𝐴} has syntax label fragment "sn" (because it is defined in df-sn 4628), and the pair construct {𝐴, 𝐵} has fragment "pr" ( from df-pr 4630). Digits are used to represent themselves. Suffixes (e.g., with numbers) are sometimes used to distinguish multiple theorems that would otherwise produce the same label.
• Phantom definitions. In some cases there are common label fragments for something that could be in a definition, but for technical reasons is not. The is-element-of (is member of) construct 𝐴 ∈ 𝐵 does not have a df-NAME definition; in this case its syntax label fragment is "el". Thus, because the theorem beginning with (𝐴 ∈ (𝐵 ∖ {𝐶}) uses is-element-of ("el") of a class difference ("dif") of a singleton ("sn"), it is labeled eldifsn 4789. An "n" is often used for negation (¬), e.g., nan 828.
• Exceptions. Sometimes there is a definition df-NAME but the label fragment is not the NAME part. The definition should note this exception as part of its definition. In addition, the table below attempts to list all such cases and marks them in bold. For example, the label fragment "cn" represents complex numbers ℂ (even though its definition is in df-c 11112) and "re" represents real numbers ℝ (Definition df-r 11116). The empty set ∅ often uses fragment 0, even though it is defined in df-nul 4322. The syntax construct (𝐴 + 𝐵) usually uses the fragment "add" (which is consistent with df-add 11117), but "p" is used as the fragment for constant theorems. Equality (𝐴 = 𝐵) often uses "e" as the fragment. As a result, "two plus two equals four" is labeled 2p2e4 12343.
• Other markings. In labels we sometimes use "com" for "commutative", "ass" for "associative", "rot" for "rotation", and "di" for "distributive".
• Focus on the important part of the conclusion. Typically the conclusion is the part the user is most interested in. So, a rough guideline is that a label typically provides a hint about only the conclusion; a label rarely says anything about the hypotheses or antecedents. If there are multiple theorems with the same conclusion but different hypotheses/antecedents, then the labels will need to differ; those label differences should emphasize what is different. There is no need to always fully describe the conclusion; just identify the important part. For example, cos0 16089 is the theorem that provides the value for the cosine of 0; we would need to look at the theorem itself to see what that value is. The label "cos0" is concise and we use it instead of "cos0eq1". There is no need to add the "eq1", because there will never be a case where we have to disambiguate between different values produced by the cosine of zero, and we generally prefer shorter labels if they are unambiguous.
• Closures and values. As noted above, if a function df-NAME is defined, there is typically a proof of its value labeled "NAMEval" and of its closure labeled "NAMEcl". E.g., for cosine (df-cos 16010) we have value cosval 16062 and closure coscl 16066.
• Special cases. Sometimes, syntax and related markings are insufficient to distinguish different theorems. For example, there are over a hundred different implication-only theorems. They are grouped in a more ad-hoc way that attempts to make their distinctions clearer. These often use abbreviations such as "mp" for "modus ponens", "syl" for syllogism, and "id" for "identity". It is especially hard to give good names in the propositional calculus section because there are so few primitives. However, in most cases this is not a serious problem. There are a few very common theorems like ax-mp 5 and syl 17 that you will have no trouble remembering, a few theorem series like syl*anc and simp* that you can use parametrically, and a few other useful glue things for destructuring 'and's and 'or's (see natded 29645 for a list), and that is about all you need for most things. As for the rest, you can just assume that if it involves at most three connectives, then it is probably already proved in set.mm, and searching for it will give you the label.
• Suffixes. Suffixes are used to indicate the form of a theorem (inference, deduction, or closed form, see above). Additionally, we sometimes suffix with "v" the label of a theorem adding a disjoint variable condition, as in 19.21v 1942 versus 19.21 2200. This often permits to prove the result using fewer axioms, and/or to eliminate a nonfreeness hypothesis (such as Ⅎ𝑥𝜑 in 19.21 2200). If no constraint is put on axiom use, then the v-version can be proved from the original theorem using nfv 1917. If two (resp. three) such disjoint variable conditions are added, then the suffix "vv" (resp. "vvv") is used, e.g., exlimivv 1935. Conversely, we sometimes suffix with "f" the label of a theorem introducing such a hypothesis to eliminate the need for the disjoint variable condition; e.g., euf 2570 derived from eu6 2568. The "f" stands for "not free in" which is less restrictive than "does not occur in." The suffix "b" often means "biconditional" (↔, "iff" , "if and only if"), e.g., sspwb 5448. We sometimes suffix with "s" the label of an inference that manipulates an antecedent, leaving the consequent unchanged. The "s" means that the inference eliminates the need for a syllogism (syl 17) -type inference in a proof. A theorem label is suffixed with "ALT" if it provides an alternate less-preferred proof of a theorem (e.g., the proof is clearer but uses more axioms than the preferred version). The "ALT" may be further suffixed with a number if there is more than one alternate theorem. Furthermore, a theorem label is suffixed with "OLD" if there is a new version of it and the OLD version is obsolete (and will be removed within one year). Finally, it should be mentioned that suffixes can be combined, for example in cbvaldva 2408 (cbval 2397 in deduction form "d" with a not free variable replaced by a disjoint variable condition "v" with a conjunction as antecedent "a"). As a general rule, the suffixes for the theorem forms ("i", "d" or "g") should be the first of multiple suffixes, as for example in vtocldf 3541. Here is a non-exhaustive list of common suffixes:
  • a : theorem having a conjunction as antecedent
  • b : theorem expressing a logical equivalence
  • c : contraction (e.g., sylc 65, syl2anc 584), commutes (e.g., biimpac 479)
  • d : theorem in deduction form
  • f : theorem with a hypothesis such as Ⅎ𝑥𝜑
  • g : theorem in closed form having an "is a set" antecedent
  • i : theorem in inference form
  • l : theorem concerning something at the left
  • r : theorem concerning something at the right
  • r : theorem with something reversed (e.g., a biconditional)
  • s : inference that manipulates an antecedent ("s" refers to an application of syl 17 that is eliminated)
  • t : theorem in closed form (not having an "is a set" antecedent)
  • v : theorem with one (main) disjoint variable condition
  • vv : theorem with two (main) disjoint variable conditions
  • w : weak(er) form of a theorem
  • ALT : alternate proof of a theorem
  • ALTV : alternate version of a theorem or definition (mathbox only)
  • OLD : old/obsolete version of a theorem (or proof) or definition
• Reuse. When creating a new theorem or axiom, try to reuse abbreviations used elsewhere. A comment should explain the first use of an abbreviation.

The following table shows some commonly used abbreviations in labels, in alphabetical order. For each abbreviation we provide a mnenomic, the source theorem or the assumption defining it, an expression showing what it looks like, whether or not it is a "syntax fragment" (an abbreviation that indicates a particular kind of syntax), and hyperlinks to label examples that use the abbreviation. The abbreviation is bolded if there is a df-NAME definition but the label fragment is not NAME. This is not a complete list of abbreviations, though we do want this to eventually be a complete list of exceptions.

Abbreviation	Mnenomic	Source	Expression	Syntax?	Example(s)
a	and (suffix)			No	biimpa 477, rexlimiva 3147
abl	Abelian group	df-abl 19645	Abel	Yes	ablgrp 19647, zringabl 21013
abs	absorption			No	ressabs 17190
abs	absolute value (of a complex number)	df-abs 15179	(abs‘𝐴)	Yes	absval 15181, absneg 15220, abs1 15240
ad	adding			No	adantr 481, ad2antlr 725
add	add (see "p")	df-add 11117	(𝐴 + 𝐵)	Yes	addcl 11188, addcom 11396, addass 11193
al	"for all"		∀𝑥𝜑	No	alim 1812, alex 1828
ALT	alternative/less preferred (suffix)			No	idALT 23
an	and	df-an 397	(𝜑 ∧ 𝜓)	Yes	anor 981, iman 402, imnan 400
ant	antecedent			No	adantr 481
ass	associative			No	biass 385, orass 920, mulass 11194
asym	asymmetric, antisymmetric			No	intasym 6113, asymref 6114, posasymb 18268
ax	axiom			No	ax6dgen 2124, ax1cn 11140
bas, base	base (set of an extensible structure)	df-base 17141	(Base‘𝑆)	Yes	baseval 17142, ressbas 17175, cnfldbas 20940
b, bi	biconditional ("iff", "if and only if")	df-bi 206	(𝜑 ↔ 𝜓)	Yes	impbid 211, sspwb 5448
br	binary relation	df-br 5148	𝐴𝑅𝐵	Yes	brab1 5195, brun 5198
c	commutes, commuted (suffix)			No	biimpac 479
c	contraction (suffix)			No	sylc 65, syl2anc 584
cbv	change bound variable			No	cbvalivw 2010, cbvrex 3359
cdm	codomain			No	ffvelcdm 7080, focdmex 7938
cl	closure			No	ifclda 4562, ovrcl 7446, zaddcl 12598
cn	complex numbers	df-c 11112	ℂ	Yes	nnsscn 12213, nncn 12216
cnfld	field of complex numbers	df-cnfld 20937	ℂfld	Yes	cnfldbas 20940, cnfldinv 20968
cntz	centralizer	df-cntz 19175	(Cntz‘𝑀)	Yes	cntzfval 19178, dprdfcntz 19879
cnv	converse	df-cnv 5683	◡𝐴	Yes	opelcnvg 5878, f1ocnv 6842
co	composition	df-co 5684	(𝐴 ∘ 𝐵)	Yes	cnvco 5883, fmptco 7123
com	commutative			No	orcom 868, bicomi 223, eqcomi 2741
con	contradiction, contraposition			No	condan 816, con2d 134
csb	class substitution	df-csb 3893	⦋𝐴 / 𝑥⦌𝐵	Yes	csbid 3905, csbie2g 3935
cyg	cyclic group	df-cyg 19740	CycGrp	Yes	iscyg 19741, zringcyg 21030
d	deduction form (suffix)			No	idd 24, impbid 211
df	(alternate) definition (prefix)			No	dfrel2 6185, dffn2 6716
di, distr	distributive			No	andi 1006, imdi 390, ordi 1004, difindi 4280, ndmovdistr 7592
dif	class difference	df-dif 3950	(𝐴 ∖ 𝐵)	Yes	difss 4130, difindi 4280
div	division	df-div 11868	(𝐴 / 𝐵)	Yes	divcl 11874, divval 11870, divmul 11871
dm	domain	df-dm 5685	dom 𝐴	Yes	dmmpt 6236, iswrddm0 14484
e, eq, equ	equals (equ for setvars, eq for classes)	df-cleq 2724	𝐴 = 𝐵	Yes	2p2e4 12343, uneqri 4150, equtr 2024
edg	edge	df-edg 28297	(Edg‘𝐺)	Yes	edgopval 28300, usgredgppr 28442
el	element of		𝐴 ∈ 𝐵	Yes	eldif 3957, eldifsn 4789, elssuni 4940
en	equinumerous	df-en	𝐴 ≈ 𝐵	Yes	domen 8953, enfi 9186
eu	"there exists exactly one"	eu6 2568	∃!𝑥𝜑	Yes	euex 2571, euabsn 4729
ex	exists (i.e. is a set)		∈ V	No	brrelex1 5727, 0ex 5306
ex, e	"there exists (at least one)"	df-ex 1782	∃𝑥𝜑	Yes	exim 1836, alex 1828
exp	export			No	expt 177, expcom 414
f	"not free in" (suffix)			No	equs45f 2458, sbf 2262
f	function	df-f 6544	𝐹:𝐴⟶𝐵	Yes	fssxp 6742, opelf 6749
fal	false	df-fal 1554	⊥	Yes	bifal 1557, falantru 1576
fi	finite intersection	df-fi 9402	(fi‘𝐵)	Yes	fival 9403, inelfi 9409
fi, fin	finite	df-fin 8939	Fin	Yes	isfi 8968, snfi 9040, onfin 9226
fld	field (Note: there is an alternative definition Fld of a field, see df-fld 36848)	df-field 20310	Field	Yes	isfld 20318, fldidom 20915
fn	function with domain	df-fn 6543	𝐴 Fn 𝐵	Yes	ffn 6714, fndm 6649
frgp	free group	df-frgp 19572	(freeGrp‘𝐼)	Yes	frgpval 19620, frgpadd 19625
fsupp	finitely supported function	df-fsupp 9358	𝑅 finSupp 𝑍	Yes	isfsupp 9361, fdmfisuppfi 9368, fsuppco 9393
fun	function	df-fun 6542	Fun 𝐹	Yes	funrel 6562, ffun 6717
fv	function value	df-fv 6548	(𝐹‘𝐴)	Yes	fvres 6907, swrdfv 14594
fz	finite set of sequential integers	df-fz 13481	(𝑀...𝑁)	Yes	fzval 13482, eluzfz 13492
fz0	finite set of sequential nonnegative integers		(0...𝑁)	Yes	nn0fz0 13595, fz0tp 13598
fzo	half-open integer range	df-fzo 13624	(𝑀..^𝑁)	Yes	elfzo 13630, elfzofz 13644
g	more general (suffix); eliminates "is a set" hypotheses			No	uniexg 7726
gr	graph			No	uhgrf 28311, isumgr 28344, usgrres1 28561
grp	group	df-grp 18818	Grp	Yes	isgrp 18821, tgpgrp 23573
gsum	group sum	df-gsum 17384	(𝐺 Σg 𝐹)	Yes	gsumval 18592, gsumwrev 19227
hash	size (of a set)	df-hash 14287	(♯‘𝐴)	Yes	hashgval 14289, hashfz1 14302, hashcl 14312
hb	hypothesis builder (prefix)			No	hbxfrbi 1827, hbald 2168, hbequid 37767
hm	(monoid, group, ring) homomorphism			No	ismhm 18669, isghm 19086, isrhm 20249
i	inference (suffix)			No	eleq1i 2824, tcsni 9734
i	implication (suffix)			No	brwdomi 9559, infeq5i 9627
id	identity			No	biid 260
iedg	indexed edge	df-iedg 28248	(iEdg‘𝐺)	Yes	iedgval0 28289, edgiedgb 28303
idm	idempotent			No	anidm 565, tpidm13 4759
im, imp	implication (label often omitted)	df-im 15044	(𝐴 → 𝐵)	Yes	iman 402, imnan 400, impbidd 209
ima	image	df-ima 5688	(𝐴 “ 𝐵)	Yes	resima 6013, imaundi 6146
imp	import			No	biimpa 477, impcom 408
in	intersection	df-in 3954	(𝐴 ∩ 𝐵)	Yes	elin 3963, incom 4200
inf	infimum	df-inf 9434	inf(ℝ+, ℝ*, < )	Yes	fiinfcl 9492, infiso 9499
is...	is (something a) ...?			No	isring 20053
j	joining, disjoining			No	jc 161, jaoi 855
l	left			No	olcd 872, simpl 483
map	mapping operation or set exponentiation	df-map 8818	(𝐴 ↑m 𝐵)	Yes	mapvalg 8826, elmapex 8838
mat	matrix	df-mat 21899	(𝑁 Mat 𝑅)	Yes	matval 21902, matring 21936
mdet	determinant (of a square matrix)	df-mdet 22078	(𝑁 maDet 𝑅)	Yes	mdetleib 22080, mdetrlin 22095
mgm	magma	df-mgm 18557	Magma	Yes	mgmidmo 18575, mgmlrid 18582, ismgm 18558
mgp	multiplicative group	df-mgp 19982	(mulGrp‘𝑅)	Yes	mgpress 19996, ringmgp 20055
mnd	monoid	df-mnd 18622	Mnd	Yes	mndass 18630, mndodcong 19404
mo	"there exists at most one"	df-mo 2534	∃*𝑥𝜑	Yes	eumo 2572, moim 2538
mp	modus ponens	ax-mp 5		No	mpd 15, mpi 20
mpo	maps-to notation for an operation	df-mpo 7410	(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)	Yes	mpompt 7518, resmpo 7524
mpt	modus ponendo tollens			No	mptnan 1770, mptxor 1771
mpt	maps-to notation for a function	df-mpt 5231	(𝑥 ∈ 𝐴 ↦ 𝐵)	Yes	fconstmpt 5736, resmpt 6035
mpt2	maps-to notation for an operation (deprecated). We are in the process of replacing mpt2 with mpo in labels.	df-mpo 7410	(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)	Yes	mpompt 7518, resmpo 7524
mul	multiplication (see "t")	df-mul 11118	(𝐴 · 𝐵)	Yes	mulcl 11190, divmul 11871, mulcom 11192, mulass 11194
n, not	not		¬ 𝜑	Yes	nan 828, notnotr 130
ne	not equal	df-ne	𝐴 ≠ 𝐵	Yes	exmidne 2950, neeqtrd 3010
nel	not element of	df-nel	𝐴 ∉ 𝐵	Yes	neli 3048, nnel 3056
ne0	not equal to zero (see n0)		≠ 0	No	negne0d 11565, ine0 11645, gt0ne0 11675
nf	"not free in" (prefix)			No	nfnd 1861
ngp	normed group	df-ngp 24083	NrmGrp	Yes	isngp 24096, ngptps 24102
nm	norm (on a group or ring)	df-nm 24082	(norm‘𝑊)	Yes	nmval 24089, subgnm 24133
nn	positive integers	df-nn 12209	ℕ	Yes	nnsscn 12213, nncn 12216
nn0	nonnegative integers	df-n0 12469	ℕ0	Yes	nnnn0 12475, nn0cn 12478
n0	not the empty set (see ne0)		≠ ∅	No	n0i 4332, vn0 4337, ssn0 4399
OLD	old, obsolete (to be removed soon)			No	19.43OLD 1886
on	ordinal number	df-on 6365	𝐴 ∈ On	Yes	elon 6370, 1on 8474 onelon 6386
op	ordered pair	df-op 4634	⟨𝐴, 𝐵⟩	Yes	dfopif 4869, opth 5475
or	or	df-or 846	(𝜑 ∨ 𝜓)	Yes	orcom 868, anor 981
ot	ordered triple	df-ot 4636	⟨𝐴, 𝐵, 𝐶⟩	Yes	euotd 5512, fnotovb 7455
ov	operation value	df-ov 7408	(𝐴𝐹𝐵)	Yes	fnotovb 7455, fnovrn 7578
p	plus (see "add"), for all-constant theorems	df-add 11117	(3 + 2) = 5	Yes	3p2e5 12359
pfx	prefix	df-pfx 14617	(𝑊 prefix 𝐿)	Yes	pfxlen 14629, ccatpfx 14647
pm	Principia Mathematica			No	pm2.27 42
pm	partial mapping (operation)	df-pm 8819	(𝐴 ↑pm 𝐵)	Yes	elpmi 8836, pmsspw 8867
pr	pair	df-pr 4630	{𝐴, 𝐵}	Yes	elpr 4650, prcom 4735, prid1g 4763, prnz 4780
prm, prime	prime (number)	df-prm 16605	ℙ	Yes	1nprm 16612, dvdsprime 16620
pss	proper subset	df-pss 3966	𝐴 ⊊ 𝐵	Yes	pssss 4094, sspsstri 4101
q	rational numbers ("quotients")	df-q 12929	ℚ	Yes	elq 12930
r	reversed (suffix)			No	pm4.71r 559, caovdir 7637
r	right			No	orcd 871, simprl 769
rab	restricted class abstraction	df-rab 3433	{𝑥 ∈ 𝐴 ∣ 𝜑}	Yes	rabswap 3441, df-oprab 7409
ral	restricted universal quantification	df-ral 3062	∀𝑥 ∈ 𝐴𝜑	Yes	ralnex 3072, ralrnmpo 7543
rcl	reverse closure			No	ndmfvrcl 6924, nnarcl 8612
re	real numbers	df-r 11116	ℝ	Yes	recn 11196, 0re 11212
rel	relation	df-rel 5682	Rel 𝐴	Yes	brrelex1 5727, relmpoopab 8076
res	restriction	df-res 5687	(𝐴 ↾ 𝐵)	Yes	opelres 5985, f1ores 6844
reu	restricted existential uniqueness	df-reu 3377	∃!𝑥 ∈ 𝐴𝜑	Yes	nfreud 3429, reurex 3380
rex	restricted existential quantification	df-rex 3071	∃𝑥 ∈ 𝐴𝜑	Yes	rexnal 3100, rexrnmpo 7544
rmo	restricted "at most one"	df-rmo 3376	∃*𝑥 ∈ 𝐴𝜑	Yes	nfrmod 3428, nrexrmo 3397
rn	range	df-rn 5686	ran 𝐴	Yes	elrng 5889, rncnvcnv 5931
ring	(unital) ring	df-ring 20051	Ring	Yes	ringidval 20000, isring 20053, ringgrp 20054
rng	non-unital ring	df-rng 46635	Rng	Yes	isrng 46636, rngabl 46637, rnglz 46650
rot	rotation			No	3anrot 1100, 3orrot 1092
s	eliminates need for syllogism (suffix)			No	ancoms 459
sb	(proper) substitution (of a set)	df-sb 2068	[𝑦 / 𝑥]𝜑	Yes	spsbe 2085, sbimi 2077
sbc	(proper) substitution of a class	df-sbc 3777	[𝐴 / 𝑥]𝜑	Yes	sbc2or 3785, sbcth 3791
sca	scalar	df-sca 17209	(Scalar‘𝐻)	Yes	resssca 17284, mgpsca 19989
simp	simple, simplification			No	simpl 483, simp3r3 1283
sn	singleton	df-sn 4628	{𝐴}	Yes	eldifsn 4789
sp	specialization			No	spsbe 2085, spei 2393
ss	subset	df-ss 3964	𝐴 ⊆ 𝐵	Yes	difss 4130
struct	structure	df-struct 17076	Struct	Yes	brstruct 17077, structfn 17085
sub	subtract	df-sub 11442	(𝐴 − 𝐵)	Yes	subval 11447, subaddi 11543
sup	supremum	df-sup 9433	sup(𝐴, 𝐵, < )	Yes	fisupcl 9460, supmo 9443
supp	support (of a function)	df-supp 8143	(𝐹 supp 𝑍)	Yes	ressuppfi 9386, mptsuppd 8168
swap	swap (two parts within a theorem)			No	rabswap 3441, 2reuswap 3741
syl	syllogism	syl 17		No	3syl 18
sym	symmetric			No	df-symdif 4241, cnvsym 6110
symg	symmetric group	df-symg 19229	(SymGrp‘𝐴)	Yes	symghash 19239, pgrpsubgsymg 19271
t	times (see "mul"), for all-constant theorems	df-mul 11118	(3 · 2) = 6	Yes	3t2e6 12374
th, t	theorem			No	nfth 1803, sbcth 3791, weth 10486, ancomst 465
tp	triple	df-tp 4632	{𝐴, 𝐵, 𝐶}	Yes	eltpi 4690, tpeq1 4745
tr	transitive			No	bitrd 278, biantr 804
tru, t	true, truth	df-tru 1544	⊤	Yes	bitru 1550, truanfal 1575, biimt 360
un	union	df-un 3952	(𝐴 ∪ 𝐵)	Yes	uneqri 4150, uncom 4152
unit	unit (in a ring)	df-unit 20164	(Unit‘𝑅)	Yes	isunit 20179, nzrunit 20293
v	setvar (especially for specializations of theorems when a class is replaced by a setvar variable)		x	Yes	cv 1540, vex 3478, velpw 4606, vtoclf 3547
v	disjoint variable condition used in place of nonfreeness hypothesis (suffix)			No	spimv 2389
vtx	vertex	df-vtx 28247	(Vtx‘𝐺)	Yes	vtxval0 28288, opvtxov 28254
vv	two disjoint variable conditions used in place of nonfreeness hypotheses (suffix)			No	19.23vv 1946
w	weak (version of a theorem) (suffix)			No	ax11w 2126, spnfw 1983
wrd	word	df-word 14461	Word 𝑆	Yes	iswrdb 14466, wrdfn 14474, ffz0iswrd 14487
xp	cross product (Cartesian product)	df-xp 5681	(𝐴 × 𝐵)	Yes	elxp 5698, opelxpi 5712, xpundi 5742
xr	eXtended reals	df-xr 11248	ℝ*	Yes	ressxr 11254, rexr 11256, 0xr 11257
z	integers (from German "Zahlen")	df-z 12555	ℤ	Yes	elz 12556, zcn 12559
zn	ring of integers mod 𝑁	df-zn 21047	(ℤ/nℤ‘𝑁)	Yes	znval 21078, zncrng 21091, znhash 21105
zring	ring of integers	df-zring 21010	ℤring	Yes	zringbas 21015, zringcrng 21011
0, z	slashed zero (empty set)	df-nul 4322	∅	Yes	n0i 4332, vn0 4337; snnz 4779, prnz 4780

(Contributed by the Metamath team, 27-Dec-2016.) Date of last revision. (Revised by the Metamath team, 22-Sep-2022.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
conventions-labels.1	⊢ 𝜑

Assertion

Ref	Expression
conventions-labels	⊢ 𝜑

Proof of Theorem conventions-labels

Step	Hyp	Ref	Expression
1		conventions-labels.1	1 ⊢ 𝜑

This theorem is referenced by: (None)