**Description: **If 𝜑 is a theorem, then any set
belongs to the class
{𝑥
∣ 𝜑}.
Therefore, {𝑥 ∣ 𝜑} is "a" universal class.
This is the closest one can get to defining a universal class, or
proving vex 3176, without using ax-ext 2590. Note that this theorem has no
dv condition and does not use df-clel 2606 nor df-cleq 2603 either: only
first-order logic and df-clab 2597.
Without ax-ext 2590, one cannot define "the" universal
class, since one
could not prove for instance the justification theorem
{𝑥
∣ ⊤} = {𝑦
∣ ⊤} (see vjust 3174). Indeed, in order to prove
any equality of classes, one needs df-cleq 2603, which has ax-ext 2590 as a
hypothesis. Therefore, the classes {𝑥 ∣ ⊤},
{𝑦
∣ (𝜑 → 𝜑)}, {𝑧 ∣ (∀𝑡𝑡 = 𝑡 → ∀𝑡𝑡 = 𝑡)} and
countless others are all universal classes whose equality one cannot
prove without ax-ext 2590. See also bj-issetw 32054.
A version with a dv condition between 𝑥 and 𝑦 and not
requiring
ax-13 2234 is proved as bj-vexwv 32051, while the degenerate instance is a
simple consequence of abid 2598. (Contributed by BJ, 13-Jun-2019.)
(Proof modification is discouraged.) Use bj-vexwv 32051 instead when
sufficient. (New usage is discouraged.) |