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Theorem bj-vexw 33181
Description: If 𝜑 is a theorem, then any set belongs to the class {𝑥𝜑}. Therefore, {𝑥𝜑} is "a" universal class.

This is the closest one can get to defining a universal class, or proving vex 3354, without using ax-ext 2751. Note that this theorem has no dv condition and does not use df-clel 2767 nor df-cleq 2764 either: only first-order logic and df-clab 2758.

Without ax-ext 2751, one cannot define "the" universal class, since one could not prove for instance the justification theorem {𝑥 ∣ ⊤} = {𝑦 ∣ ⊤} (see vjust 3352). Indeed, in order to prove any equality of classes, one needs df-cleq 2764, which has ax-ext 2751 as a hypothesis. Therefore, the classes {𝑥 ∣ ⊤}, {𝑦 ∣ (𝜑𝜑)}, {𝑧 ∣ (∀𝑡𝑡 = 𝑡 → ∀𝑡𝑡 = 𝑡)} and countless others are all universal classes whose equality one cannot prove without ax-ext 2751. See also bj-issetw 33186.

A version with a dv condition between 𝑥 and 𝑦 and not requiring ax-13 2408 is proved as bj-vexwv 33183, while the degenerate instance is a simple consequence of abid 2759. (Contributed by BJ, 13-Jun-2019.) (Proof modification is discouraged.) Use bj-vexwv 33183 instead when sufficient. (New usage is discouraged.)

Hypothesis
Ref Expression
bj-vexw.1 𝜑
Assertion
Ref Expression
bj-vexw 𝑦 ∈ {𝑥𝜑}

Proof of Theorem bj-vexw
StepHypRef Expression
1 bj-vexwt 33180 . 2 (∀𝑥𝜑𝑦 ∈ {𝑥𝜑})
2 bj-vexw.1 . 2 𝜑
31, 2mpg 1872 1 𝑦 ∈ {𝑥𝜑}
Colors of variables: wff setvar class
Syntax hints:  wcel 2145  {cab 2757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-12 2203  ax-13 2408
This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853  df-sb 2050  df-clab 2758
This theorem is referenced by:  bj-ralvw  33191
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