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Theorem bj-df-v 33375
Description: Alternate definition of the universal class. Actually, the current definition df-v 3352 should be proved from this one, and vex 3353 should be proved from this proposed definition together with bj-vexwv 33213, which would remove from vex 3353 dependency on ax-13 2352 (see also comment of bj-vexw 33211). (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 2759 . 2 (V = {𝑥 ∣ ⊤} ↔ ∀𝑦(𝑦 ∈ V ↔ 𝑦 ∈ {𝑥 ∣ ⊤}))
2 vex 3353 . . 3 𝑦 ∈ V
3 tru 1657 . . . 4
43bj-vexwv 33213 . . 3 𝑦 ∈ {𝑥 ∣ ⊤}
52, 42th 255 . 2 (𝑦 ∈ V ↔ 𝑦 ∈ {𝑥 ∣ ⊤})
61, 5mpgbir 1894 1 V = {𝑥 ∣ ⊤}
Colors of variables: wff setvar class
Syntax hints:  wb 197   = wceq 1652  wtru 1653  wcel 2155  {cab 2751  Vcvv 3350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-tru 1656  df-ex 1875  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-v 3352
This theorem is referenced by: (None)
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