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Theorem bj-vexw 32049
 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.)
Hypothesis
Ref Expression
bj-vexw.1 𝜑
Assertion
Ref Expression
bj-vexw 𝑦 ∈ {𝑥𝜑}

Proof of Theorem bj-vexw
StepHypRef Expression
1 bj-vexwt 32048 . 2 (∀𝑥𝜑𝑦 ∈ {𝑥𝜑})
2 bj-vexw.1 . 2 𝜑
31, 2mpg 1715 1 𝑦 ∈ {𝑥𝜑}
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 1977  {cab 2596 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034  ax-13 2234 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-sb 1868  df-clab 2597 This theorem is referenced by:  bj-ralvw  32059
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