Theorem bj-df-v 34471

Description: Alternate definition of the universal class. Actually, the current definition df-v 3443 should be proved from this one, and vex 3444 should be proved from this proposed definition together with vexw 2782, which would remove from vex 3444 dependency on ax-13 2379 (see also comment of vexw 2782). (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 2792	. 2 ⊢ (V = {𝑥 ∣ ⊤} ↔ ∀𝑦(𝑦 ∈ V ↔ 𝑦 ∈ {𝑥 ∣ ⊤}))
2		vex 3444	. . 3 ⊢ 𝑦 ∈ V
3		tru 1542	. . . 4 ⊢ ⊤
4	3	vexw 2782	. . 3 ⊢ 𝑦 ∈ {𝑥 ∣ ⊤}
5	2, 4	2th 267	. 2 ⊢ (𝑦 ∈ V ↔ 𝑦 ∈ {𝑥 ∣ ⊤})
6	1, 5	mpgbir 1801	1 ⊢ V = {𝑥 ∣ ⊤}

Syntax hints:   ↔ wb 209   = wceq 1538  ⊤wtru 1539   ∈ wcel 2111  {cab 2776  Vcvv 3441

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443

This theorem is referenced by: (None)