Theorem dmep 5917

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 3464	. 2 ⊢ (dom E = V ↔ ∀𝑥 𝑥 ∈ dom E )
2		el 5413	. . . 4 ⊢ ∃𝑦 𝑥 ∈ 𝑦
3		epel 5561	. . . . 5 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
4	3	exbii 1881	. . . 4 ⊢ (∃𝑦 𝑥 E 𝑦 ↔ ∃𝑦 𝑥 ∈ 𝑦)
5	2, 4	mpbir 234	. . 3 ⊢ ∃𝑦 𝑥 E 𝑦
6		vex 3458	. . . 4 ⊢ 𝑥 ∈ V
7	6	eldm 5894	. . 3 ⊢ (𝑥 ∈ dom E ↔ ∃𝑦 𝑥 E 𝑦)
8	5, 7	mpbir 234	. 2 ⊢ 𝑥 ∈ dom E
9	1, 8	mpgbir 1832	1 ⊢ dom E = V

This proof depends on syntax axioms:   = wceq 1570  ∃wex 1812   ∈ wcel 2147  Vcvv 3454   class class class wbr 5108   E cep 5557  dom cdm 5658

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-eprel 5558  df-dm 5668

This theorem is used by: (None)