| Description: Define the subclass
relationship. Definition 5.9 of [TakeutiZaring]
p. 17. For example, {1, 2} ⊆ {1, 2, 3}
(ex-ss 30756). Note
that 𝐴 ⊆ 𝐴 (proved in ssid 3960). Contrast this relationship with
the relationship 𝐴 ⊊ 𝐵 (as will be defined in df-pss 3926). For an
alternative definition, not requiring a dummy variable, see dfss2 3924.
Other possible definitions are given by dfss3 3927, dfss4 4223, sspss 4057,
ssequn1 4140, ssequn2 4143, sseqin2 4177, and ssdif0 4322.
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 11109
and 1ex 11204) or ℝ ( cf. df-r 11111
and
reex 11192). 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 11192). 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 3924. (Revised by GG,
15-May-2025.) |
