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Theorem bj-df-v 32207
 Description: Alternate definition of the universal class. Actually, the current definition df-v 3175 should be proved from this one, and vex 3176 should be proved from this proposed definition together with bj-vexwv 32051, which would remove from vex 3176 dependency on ax-13 2234 (see also comment of bj-vexw 32049). (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-df-v V = {𝑥 ∣ ⊤}

Proof of Theorem bj-df-v
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfcleq 2604 . 2 (V = {𝑥 ∣ ⊤} ↔ ∀𝑦(𝑦 ∈ V ↔ 𝑦 ∈ {𝑥 ∣ ⊤}))
2 vex 3176 . . 3 𝑦 ∈ V
3 tru 1479 . . . 4
43bj-vexwv 32051 . . 3 𝑦 ∈ {𝑥 ∣ ⊤}
52, 42th 253 . 2 (𝑦 ∈ V ↔ 𝑦 ∈ {𝑥 ∣ ⊤})
61, 5mpgbir 1717 1 V = {𝑥 ∣ ⊤}
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475  ⊤wtru 1476   ∈ wcel 1977  {cab 2596  Vcvv 3173 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-ext 2590 This theorem depends on definitions:  df-bi 196  df-an 385  df-tru 1478  df-ex 1696  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175 This theorem is referenced by: (None)
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