Theorem dmep 5895

Description: The domain of the membership relation is the universal class. (Contributed by Scott Fenton, 27-Oct-2010.) (Proof shortened by BJ, 26-Dec-2023.)

Assertion

Ref	Expression
dmep	⊢ dom E = V

Proof of Theorem dmep

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqv 3463	. 2 ⊢ (dom E = V ↔ ∀𝑥 𝑥 ∈ dom E )
2		el 5402	. . . 4 ⊢ ∃𝑦 𝑥 ∈ 𝑦
3		epel 5546	. . . . 5 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
4	3	exbii 1867	. . . 4 ⊢ (∃𝑦 𝑥 E 𝑦 ↔ ∃𝑦 𝑥 ∈ 𝑦)
5	2, 4	mpbir 233	. . 3 ⊢ ∃𝑦 𝑥 E 𝑦
6		vex 3457	. . . 4 ⊢ 𝑥 ∈ V
7	6	eldm 5872	. . 3 ⊢ (𝑥 ∈ dom E ↔ ∃𝑦 𝑥 E 𝑦)
8	5, 7	mpbir 233	. 2 ⊢ 𝑥 ∈ dom E
9	1, 8	mpgbir 1818	1 ⊢ dom E = V

Syntax hints:   = wceq 1559  ∃wex 1798   ∈ wcel 2141  Vcvv 3453   class class class wbr 5097   E cep 5542  dom cdm 5643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5243  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-eprel 5543  df-dm 5653

This theorem is referenced by: (None)