Theorem bj-df-v 34341

Description: Alternate definition of the universal class. Actually, the current definition df-v 3497 should be proved from this one, and vex 3498 should be proved from this proposed definition together with vexw 2805, which would remove from vex 3498 dependency on ax-13 2386 (see also comment of vexw 2805). (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.

Step	Hyp	Ref	Expression
1		dfcleq 2815	. 2 ⊢ (V = {𝑥 ∣ ⊤} ↔ ∀𝑦(𝑦 ∈ V ↔ 𝑦 ∈ {𝑥 ∣ ⊤}))
2		vex 3498	. . 3 ⊢ 𝑦 ∈ V
3		tru 1537	. . . 4 ⊢ ⊤
4	3	vexw 2805	. . 3 ⊢ 𝑦 ∈ {𝑥 ∣ ⊤}
5	2, 4	2th 266	. 2 ⊢ (𝑦 ∈ V ↔ 𝑦 ∈ {𝑥 ∣ ⊤})
6	1, 5	mpgbir 1796	1 ⊢ V = {𝑥 ∣ ⊤}

Syntax hints:   ↔ wb 208   = wceq 1533  ⊤wtru 1534   ∈ wcel 2110  {cab 2799  Vcvv 3495

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-tru 1536  df-ex 1777  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-v 3497

This theorem is referenced by: (None)