Theorem bj-df-v 33016

Description: Alternate definition of the universal class. Actually, the current definition df-v 3202 should be proved from this one, and vex 3203 should be proved from this proposed definition together with bj-vexwv 32857, which would remove from vex 3203 dependency on ax-13 2246 (see also comment of bj-vexw 32855). (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 2616	. 2 ⊢ (V = {𝑥 ∣ ⊤} ↔ ∀𝑦(𝑦 ∈ V ↔ 𝑦 ∈ {𝑥 ∣ ⊤}))
2		vex 3203	. . 3 ⊢ 𝑦 ∈ V
3		tru 1487	. . . 4 ⊢ ⊤
4	3	bj-vexwv 32857	. . 3 ⊢ 𝑦 ∈ {𝑥 ∣ ⊤}
5	2, 4	2th 254	. 2 ⊢ (𝑦 ∈ V ↔ 𝑦 ∈ {𝑥 ∣ ⊤})
6	1, 5	mpgbir 1726	1 ⊢ V = {𝑥 ∣ ⊤}

Syntax hints:   ↔ wb 196   = wceq 1483  ⊤wtru 1484   ∈ wcel 1990  {cab 2608  Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-tru 1486  df-ex 1705  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202

This theorem is referenced by: (None)