Theorem dmep 5872

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 3442	. 2 ⊢ (dom E = V ↔ ∀𝑥 𝑥 ∈ dom E )
2		el 5384	. . . 4 ⊢ ∃𝑦 𝑥 ∈ 𝑦
3		epel 5528	. . . . 5 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
4	3	exbii 1855	. . . 4 ⊢ (∃𝑦 𝑥 E 𝑦 ↔ ∃𝑦 𝑥 ∈ 𝑦)
5	2, 4	mpbir 232	. . 3 ⊢ ∃𝑦 𝑥 E 𝑦
6		vex 3436	. . . 4 ⊢ 𝑥 ∈ V
7	6	eldm 5849	. . 3 ⊢ (𝑥 ∈ dom E ↔ ∃𝑦 𝑥 E 𝑦)
8	5, 7	mpbir 232	. 2 ⊢ 𝑥 ∈ dom E
9	1, 8	mpgbir 1806	1 ⊢ dom E = V

Syntax hints:   = wceq 1547  ∃wex 1786   ∈ wcel 2119  Vcvv 3432   class class class wbr 5079   E cep 5524  dom cdm 5625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-eprel 5525  df-dm 5635

This theorem is referenced by: (None)