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Theorem bj-df-v 33322
Description: Alternate definition of the universal class. Actually, the current definition df-v 3342 should be proved from this one, and vex 3343 should be proved from this proposed definition together with bj-vexwv 33163, which would remove from vex 3343 dependency on ax-13 2391 (see also comment of bj-vexw 33161). (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 2754 . 2 (V = {𝑥 ∣ ⊤} ↔ ∀𝑦(𝑦 ∈ V ↔ 𝑦 ∈ {𝑥 ∣ ⊤}))
2 vex 3343 . . 3 𝑦 ∈ V
3 tru 1636 . . . 4
43bj-vexwv 33163 . . 3 𝑦 ∈ {𝑥 ∣ ⊤}
52, 42th 254 . 2 (𝑦 ∈ V ↔ 𝑦 ∈ {𝑥 ∣ ⊤})
61, 5mpgbir 1875 1 V = {𝑥 ∣ ⊤}
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1632  wtru 1633  wcel 2139  {cab 2746  Vcvv 3340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-12 2196  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-an 385  df-tru 1635  df-ex 1854  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-v 3342
This theorem is referenced by: (None)
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