| Description: Define the subclass
relationship. Definition 5.9 of [TakeutiZaring]
p. 17. For example, {1, 2} ⊆ {1, 2, 3}
(ex-ss 30522). Note
that 𝐴 ⊆ 𝐴 (proved in ssid 3944). Contrast this relationship with
the relationship 𝐴 ⊊ 𝐵 (as will be defined in df-pss 3910). For an
alternative definition, not requiring a dummy variable, see dfss2 3908.
Other possible definitions are given by dfss3 3911, dfss4 4204, sspss 4040,
ssequn1 4122, ssequn2 4125, sseqin2 4159, and ssdif0 4301.
We prefer the label "ss" ("subset") for ⊆, despite the fact that
it applies to classes. It is much more common to refer to this as the
subset relation than subclass, especially since most of the time the
arguments are in fact sets (and for pragmatic reasons we don't want to
need to use different operations for sets). The way set.mm is set up,
many things are technically classes despite morally (and provably) being
sets, like 1 (cf. df-1 11044
and 1ex 11138) or ℝ ( cf. df-r 11046
and
reex 11127). This has to do with the fact that there
are no "set
expressions": classes are expressions but there are only set
variables
in set.mm (cf.
https://us.metamath.org/downloads/grammar-ambiguity.txt 11127). This is
why we use ⊆ both for subclass relations
and for subset relations
and call it "subset". (Contributed by NM, 8-Jan-2002.)
Revised from
the original definition dfss2 3908. (Revised by GG,
15-May-2025.) |
