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Theorem bj-df-v 34247
 Description: Alternate definition of the universal class. Actually, the current definition df-v 3502 should be proved from this one, and vex 3503 should be proved from this proposed definition together with vexw 2810, which would remove from vex 3503 dependency on ax-13 2385 (see also comment of vexw 2810). (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 2820 . 2 (V = {𝑥 ∣ ⊤} ↔ ∀𝑦(𝑦 ∈ V ↔ 𝑦 ∈ {𝑥 ∣ ⊤}))
2 vex 3503 . . 3 𝑦 ∈ V
3 tru 1534 . . . 4
43vexw 2810 . . 3 𝑦 ∈ {𝑥 ∣ ⊤}
52, 42th 265 . 2 (𝑦 ∈ V ↔ 𝑦 ∈ {𝑥 ∣ ⊤})
61, 5mpgbir 1793 1 V = {𝑥 ∣ ⊤}
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 207   = wceq 1530  ⊤wtru 1531   ∈ wcel 2107  {cab 2804  Vcvv 3500 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-ext 2798 This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1533  df-ex 1774  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-v 3502 This theorem is referenced by: (None)
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