| Description:
The following explains some of the label conventions in use
in the Metamath Proof Explorer ("set.mm").
For the general conventions, see conventions 27258.
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 2922"rgen.1 $e |- ( x e. A -> ph ) $."
or letters corresponding to the (main) class variable used in the
hypothesis, e.g. for mdet0 20412: "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 2563 and stirling 40306.
- 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 1766, and the restricted quantifier version of Theorem 21 from
Section 19 of [Margaris] p. 90 is labeled r19.21 2956.
- 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... 14612. Furthermore, the underscore "_" should not be used.
- 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
  ![\](setminus.gif "\")   is defined in
df-dif 3577, and thus its syntax label fragment is "dif". Similarly, the
subclass relation  has syntax label fragment "ss"
because it is defined in df-ss 3588. Most theorem names follow from
these fragments, for example, the theorem proving
involves a class difference ("dif") of a subset ("ss"), and thus is
labeled difss 3737. There are many other syntax label fragments, e.g.,
singleton construct   has syntax label fragment "sn" (because it
is defined in df-sn 4178), and the pair construct  has
fragment "pr" ( from df-pr 4180). 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
 does not have a df-NAME definition;
in this case its syntax label fragment is "el". Thus, because the
theorem beginning with  uses is-element-of
("el") of a class difference ("dif") of a singleton ("sn"), it is
labeled eldifsn 4317. An "n" is often used for negation ( ), e.g.,
nan 604.
- 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
 (even though its definition is in
df-c 9942) and "re" represents real numbers  ( definition df-r 9946).
The empty set  often uses fragment 0, even though it is defined
in df-nul 3916. The syntax construct  usually uses the
fragment "add" (which is consistent with df-add 9947), but "p" is used as
the fragment for constant theorems. Equality  often uses
"e" as the fragment. As a result, "two plus two equals four" is labeled
2p2e4 11144.
- 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 14880 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 labeld "NAMEcl".
E.g., for cosine (df-cos 14801) we have value cosval 14853 and closure
coscl 14857.
- 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 27260 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 (see above).
Additionally, we sometimes suffix with "v" the label of a theorem
eliminating a hypothesis such as
   in 19.21 2075 via the use of
disjoint variable conditions combined with nfv 1843. If two (or three)
such hypotheses are eliminated, the suffix "vv" resp. "vvv" is used,
e.g. exlimivv 1860.
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 2478 derived from df-eu 2474. 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 4917.
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 2281 (cbval 2271 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 3256 or rabeqif 3191.
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 693), commutes
(e.g., biimpac 503)
- d : theorem in deduction form
- f : theorem with a hypothesis such as
- 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)
- 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
- OLD : old/obsolete version of a theorem/definition/proof
- 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 501, rexlimiva 3028 |
| abl | Abelian group | df-abl 18196 |
 | Yes | ablgrp 18198, zringabl 19822 |
| abs | absorption | | | No |
ressabs 15939 |
| abs | absolute value (of a complex number) |
df-abs 13976 |   | Yes |
absval 13978, absneg 14017, abs1 14037 |
| ad | adding | |
| No | adantr 481, ad2antlr 763 |
| add | add (see "p") | df-add 9947 |
| Yes |
addcl 10018, addcom 10222, addass 10023 |
| al | "for all" | |
 | No | alim 1738, alex 1753 |
| ALT | alternative/less preferred (suffix) | |
| No | idALT 23 |
| an | and | df-an 386 |
![/\](wedge.gif "/\")  | Yes |
anor 510, iman 440, imnan 438 |
| ant | antecedent | |
| No | adantr 481 |
| ass | associative | |
| No | biass 374, orass 546, mulass 10024 |
| asym | asymmetric, antisymmetric | |
| No | intasym 5511, asymref 5512, posasymb 16952 |
| ax | axiom | |
| No | ax6dgen 2005, ax1cn 9970 |
| bas, base |
base (set of an extensible structure) | df-base 15863 |
  | Yes |
baseval 15918, ressbas 15930, cnfldbas 19750 |
| b, bi | biconditional ("iff", "if and only if")
| df-bi 197 | | Yes |
impbid 202, sspwb 4917 |
| br | binary relation | df-br 4654 |
 | Yes | brab1 4700, brun 4703 |
| cbv | change bound variable | | |
No | cbvalivw 1934, cbvrex 3168 |
| cl | closure | | | No |
ifclda 4120, ovrcl 6686, zaddcl 11417 |
| cn | complex numbers | df-c 9942 |
| Yes | nnsscn 11025, nncn 11028 |
| cnfld | field of complex numbers | df-cnfld 19747 |
ℂfld | Yes | cnfldbas 19750, cnfldinv 19777 |
| cntz | centralizer | df-cntz 17750 |
Cntz | Yes |
cntzfval 17753, dprdfcntz 18414 |
| cnv | converse | df-cnv 5122 |
 | Yes | opelcnvg 5302, f1ocnv 6149 |
| co | composition | df-co 5123 |
 | Yes | cnvco 5308, fmptco 6396 |
| com | commutative | |
| No | orcom 402, bicomi 214, eqcomi 2631 |
| con | contradiction, contraposition | |
| No | condan 835, con2d 129 |
| csb | class substitution | df-csb 3534 |
  ![]_](_urbrack.gif "]_") | Yes |
csbid 3541, csbie2g 3564 |
| cyg | cyclic group | df-cyg 18280 |
CycGrp | Yes |
iscyg 18281, zringcyg 19839 |
| d | deduction form (suffix) | |
| No | idd 24, impbid 202 |
| df | (alternate) definition (prefix) | |
| No | dfrel2 5583, dffn2 6047 |
| di, distr | distributive | |
| No |
andi 911, imdi 378, ordi 908, difindi 3881, ndmovdistr 6823 |
| dif | class difference | df-dif 3577 |
| Yes |
difss 3737, difindi 3881 |
| div | division | df-div 10685 |
| Yes |
divcl 10691, divval 10687, divmul 10688 |
| dm | domain | df-dm 5124 |
 | Yes | dmmpt 5630, iswrddm0 13329 |
| e, eq, equ | equals | df-cleq 2615 |
| Yes |
2p2e4 11144, uneqri 3755, equtr 1948 |
| edg | edge | df-edg 25940 |
Edg | Yes |
edgopval 25944, usgredgppr 26088 |
| el | element of | |
| Yes |
eldif 3584, eldifsn 4317, elssuni 4467 |
| eu | "there exists exactly one" | df-eu 2474 |
 | Yes | euex 2494, euabsn 4261 |
| ex | exists (i.e. is a set) | |
| No | brrelex 5156, 0ex 4790 |
| ex | "there exists (at least one)" | df-ex 1705 |
 | Yes | exim 1761, alex 1753 |
| exp | export | |
| No | expt 168, expcom 451 |
| f | "not free in" (suffix) | |
| No | equs45f 2350, sbf 2380 |
| f | function | df-f 5892 |
   | Yes | fssxp 6060, opelf 6065 |
| fal | false | df-fal 1489 |
 | Yes | bifal 1497, falantru 1508 |
| fi | finite intersection | df-fi 8317 |
 | Yes | fival 8318, inelfi 8324 |
| fi, fin | finite | df-fin 7959 |
 | Yes |
isfi 7979, snfi 8038, onfin 8151 |
| fld | field (Note: there is an alternative
definition  of a field, see df-fld 33791) | df-field 18750 |
Field | Yes | isfld 18756, fldidom 19305 |
| fn | function with domain | df-fn 5891 |
 | Yes | ffn 6045, fndm 5990 |
| frgp | free group | df-frgp 18123 |
freeGrp | Yes |
frgpval 18171, frgpadd 18176 |
| fsupp | finitely supported function |
df-fsupp 8276 | finSupp  | Yes |
isfsupp 8279, fdmfisuppfi 8284, fsuppco 8307 |
| fun | function | df-fun 5890 |
 | Yes | funrel 5905, ffun 6048 |
| fv | function value | df-fv 5896 |
| Yes | fvres 6207, swrdfv 13424 |
| fz | finite set of sequential integers |
df-fz 12327 |
  | Yes | fzval 12328, eluzfz 12337 |
| fz0 | finite set of sequential nonnegative integers |
|
 | Yes | nn0fz0 12437, fz0tp 12440 |
| fzo | half-open integer range | df-fzo 12466 |
..^ | Yes |
elfzo 12472, elfzofz 12485 |
| g | more general (suffix); eliminates "is a set"
hypothsis | |
| No | uniexg 6955 |
| gr | graph | |
| No | uhgrf 25957, isumgr 25990, usgrres1 26207 |
| grp | group | df-grp 17425 |
 | Yes | isgrp 17428, tgpgrp 21882 |
| gsum | group sum | df-gsum 16103 |
 g | Yes |
gsumval 17271, gsumwrev 17796 |
| hash | size (of a set) | df-hash 13118 |
 | Yes |
hashgval 13120, hashfz1 13134, hashcl 13147 |
| hb | hypothesis builder (prefix) | |
| No | hbxfrbi 1752, hbald 2041, hbequid 34194 |
| hm | (monoid, group, ring) homomorphism | |
| No | ismhm 17337, isghm 17660, isrhm 18721 |
| i | inference (suffix) | |
| No | eleq1i 2692, tcsni 8619 |
| i | implication (suffix) | |
| No | brwdomi 8473, infeq5i 8533 |
| id | identity | |
| No | biid 251 |
| iedg | indexed edge | df-iedg 25877 |
iEdg | Yes |
iedgval0 25932, edgiedgb 25947 |
| idm | idempotent | |
| No | anidm 676, tpidm13 4291 |
| im, imp | implication (label often omitted) |
df-im 13841 |  | Yes |
iman 440, imnan 438, impbidd 200 |
| ima | image | df-ima 5127 |
 | Yes | resima 5431, imaundi 5545 |
| imp | import | |
| No | biimpa 501, impcom 446 |
| in | intersection | df-in 3581 |
 | Yes | elin 3796, incom 3805 |
| inf | infimum | df-inf 8349 |
inf   | Yes |
fiinfcl 8407, infiso 8413 |
| is... | is (something a) ...? | |
| No | isring 18551 |
| j | joining, disjoining | |
| No | jc 159, jaoi 394 |
| l | left | |
| No | olcd 408, simpl 473 |
| map | mapping operation or set exponentiation |
df-map 7859 |  | Yes |
mapvalg 7867, elmapex 7878 |
| mat | matrix | df-mat 20214 |
Mat | Yes |
matval 20217, matring 20249 |
| mdet | determinant (of a square matrix) |
df-mdet 20391 | maDet | Yes |
mdetleib 20393, mdetrlin 20408 |
| mgm | magma | df-mgm 17242 |
 | Yes |
mgmidmo 17259, mgmlrid 17266, ismgm 17243 |
| mgp | multiplicative group | df-mgp 18490 |
mulGrp | Yes |
mgpress 18500, ringmgp 18553 |
| mnd | monoid | df-mnd 17295 |
 | Yes | mndass 17302, mndodcong 17961 |
| mo | "there exists at most one" | df-mo 2475 |
 | Yes | eumo 2499, moim 2519 |
| mp | modus ponens | ax-mp 5 |
| No | mpd 15, mpi 20 |
| mpt | modus ponendo tollens | |
| No | mptnan 1693, mptxor 1694 |
| mpt | maps-to notation for a function |
df-mpt 4730 |  | Yes |
fconstmpt 5163, resmpt 5449 |
| mpt2 | maps-to notation for an operation |
df-mpt2 6655 |  | Yes |
mpt2mpt 6752, resmpt2 6758 |
| mul | multiplication (see "t") | df-mul 9948 |
 | Yes |
mulcl 10020, divmul 10688, mulcom 10022, mulass 10024 |
| n, not | not | |
| Yes |
nan 604, notnotr 125 |
| ne | not equal | df-ne |  |
Yes | exmidne 2804, neeqtrd 2863 |
| nel | not element of | df-nel | 
|
Yes | neli 2899, nnel 2906 |
| ne0 | not equal to zero (see n0) | |
| No |
negne0d 10390, ine0 10465, gt0ne0 10493 |
| nf | "not free in" (prefix) | |
| No | nfnd 1785 |
| ngp | normed group | df-ngp 22388 |
NrmGrp | Yes | isngp 22400, ngptps 22406 |
| nm | norm (on a group or ring) | df-nm 22387 |
  | Yes |
nmval 22394, subgnm 22437 |
| nn | positive integers | df-nn 11021 |
 | Yes | nnsscn 11025, nncn 11028 |
| nn0 | nonnegative integers | df-n0 11293 |
 | Yes | nnnn0 11299, nn0cn 11302 |
| n0 | not the empty set (see ne0) | |
| No | n0i 3920, vn0 3924, ssn0 3976 |
| OLD | old, obsolete (to be removed soon) | |
| No | 19.43OLD 1811 |
| op | ordered pair | df-op 4184 |
  | Yes | dfopif 4399, opth 4945 |
| or | or | df-or 385 |
 | Yes |
orcom 402, anor 510 |
| ot | ordered triple | df-ot 4186 |
| Yes |
euotd 4975, fnotovb 6694 |
| ov | operation value | df-ov 6653 |
| Yes
| fnotovb 6694, fnovrn 6809 |
| p | plus (see "add"), for all-constant
theorems | df-add 9947 |
   | Yes |
3p2e5 11160 |
| pfx | prefix | df-pfx 41382 |
prefix  | Yes |
pfxlen 41391, ccatpfx 41409 |
| pm | Principia Mathematica | |
| No | pm2.27 42 |
| pm | partial mapping (operation) | df-pm 7860 |
 | Yes | elpmi 7876, pmsspw 7892 |
| pr | pair | df-pr 4180 |
| Yes |
elpr 4198, prcom 4267, prid1g 4295, prnz 4310 |
| prm, prime | prime (number) | df-prm 15386 |
 | Yes | 1nprm 15392, dvdsprime 15400 |
| pss | proper subset | df-pss 3590 |
 | Yes | pssss 3702, sspsstri 3709 |
| q | rational numbers ("quotients") | df-q 11789 |
 | Yes | elq 11790 |
| r | right | |
| No | orcd 407, simprl 794 |
| rab | restricted class abstraction |
df-rab 2921 |  | Yes |
rabswap 3121, df-oprab 6654 |
| ral | restricted universal quantification |
df-ral 2917 | | Yes |
ralnex 2992, ralrnmpt2 6775 |
| rcl | reverse closure | |
| No | ndmfvrcl 6219, nnarcl 7696 |
| re | real numbers | df-r 9946 |
| Yes | recn 10026, 0re 10040 |
| rel | relation | df-rel 5121 |  |
Yes | brrelex 5156, relmpt2opab 7259 |
| res | restriction | df-res 5126 |
 | Yes |
opelres 5401, f1ores 6151 |
| reu | restricted existential uniqueness |
df-reu 2919 | | Yes |
nfreud 3112, reurex 3160 |
| rex | restricted existential quantification |
df-rex 2918 | | Yes |
rexnal 2995, rexrnmpt2 6776 |
| rmo | restricted "at most one" |
df-rmo 2920 | | Yes |
nfrmod 3113, nrexrmo 3163 |
| rn | range | df-rn 5125 |  |
Yes | elrng 5314, rncnvcnv 5349 |
| rng | (unital) ring | df-ring 18549 |
 | Yes |
ringidval 18503, isring 18551, ringgrp 18552 |
| rot | rotation | |
| No | 3anrot 1043, 3orrot 1044 |
| s | eliminates need for syllogism (suffix) |
| | No | ancoms 469 |
| sb | (proper) substitution (of a set) |
df-sb 1881 |  ![]](rbrack.gif "]") | Yes |
spsbe 1884, sbimi 1886 |
| sbc | (proper) substitution of a class |
df-sbc 3436 |  ![].](_drbrack.gif "].") | Yes |
sbc2or 3444, sbcth 3450 |
| sca | scalar | df-sca 15957 |
Scalar | Yes |
resssca 16031, mgpsca 18496 |
| simp | simple, simplification | |
| No | simpl 473, simp3r3 1171 |
| sn | singleton | df-sn 4178 |
| Yes | eldifsn 4317 |
| sp | specialization | |
| No | spsbe 1884, spei 2261 |
| ss | subset | df-ss 3588 |
| Yes | difss 3737 |
| struct | structure | df-struct 15859 |
Struct | Yes | brstruct 15866, structfn 15874 |
| sub | subtract | df-sub 10268 |
 | Yes |
subval 10272, subaddi 10368 |
| sup | supremum | df-sup 8348 |
 | Yes |
fisupcl 8375, supmo 8358 |
| supp | support (of a function) | df-supp 7296 |
supp | Yes |
ressuppfi 8301, mptsuppd 7318 |
| swap | swap (two parts within a theorem) |
| | No | rabswap 3121, 2reuswap 3410 |
| syl | syllogism | syl 17 |
| No | 3syl 18 |
| sym | symmetric | |
| No | df-symdif 3844, cnvsym 5510 |
| symg | symmetric group | df-symg 17798 |
 | Yes |
symghash 17805, pgrpsubgsymg 17828 |
| t |
times (see "mul"), for all-constant theorems |
df-mul 9948 |
 | Yes |
3t2e6 11179 |
| th | theorem | |
| No | nfth 1727, sbcth 3450, weth 9317 |
| tp | triple | df-tp 4182 |
| Yes |
eltpi 4229, tpeq1 4277 |
| tr | transitive | |
| No | bitrd 268, biantr 972 |
| tru | true | df-tru 1486 |
 | Yes | bitru 1496, truanfal 1507 |
| un | union | df-un 3579 |
 | Yes |
uneqri 3755, uncom 3757 |
| unit | unit (in a ring) |
df-unit 18642 | Unit | Yes |
isunit 18657, nzrunit 19267 |
| v | disjoint variable conditions used when
a not-free hypothesis (suffix) |
| | No | spimv 2257 |
| vtx | vertex | df-vtx 25876 |
Vtx | Yes |
vtxval0 25931, opvtxov 25885 |
| vv | 2 disjoint variables (in a not-free hypothesis)
(suffix) | | | No | 19.23vv 1903 |
| w | weak (version of a theorem) (suffix) | |
| No | ax11w 2007, spnfw 1928 |
| wrd | word |
df-word 13299 | Word | Yes |
iswrdb 13311, wrdfn 13319, ffz0iswrd 13332 |
| xp | cross product (Cartesian product) |
df-xp 5120 |  | Yes |
elxp 5131, opelxpi 5148, xpundi 5171 |
| xr | eXtended reals | df-xr 10078 |
| Yes | ressxr 10083, rexr 10085, 0xr 10086 |
| z | integers (from German "Zahlen") |
df-z 11378 |  | Yes |
elz 11379, zcn 11382 |
| zn | ring of integers   | df-zn 19855 |
ℤ/nℤ | Yes |
znval 19883, zncrng 19893, znhash 19907 |
| zring | ring of integers | df-zring 19819 |
ℤring | Yes | zringbas 19824, zringcrng 19820
|
| 0, z |
slashed zero (empty set) (see n0) | df-nul 3916 |
| Yes |
n0i 3920, vn0 3924; snnz 4309, prnz 4310 |
(Contributed by DAW, 27-Dec-2016.) (New usage is
discouraged.) |
