Theorem dmep 5904

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 3460	. 2 ⊢ (dom E = V ↔ ∀𝑥 𝑥 ∈ dom E )
2		el 5405	. . . 4 ⊢ ∃𝑦 𝑥 ∈ 𝑦
3		epel 5553	. . . . 5 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
4	3	exbii 1880	. . . 4 ⊢ (∃𝑦 𝑥 E 𝑦 ↔ ∃𝑦 𝑥 ∈ 𝑦)
5	2, 4	mpbir 234	. . 3 ⊢ ∃𝑦 𝑥 E 𝑦
6		vex 3454	. . . 4 ⊢ 𝑥 ∈ V
7	6	eldm 5881	. . 3 ⊢ (𝑥 ∈ dom E ↔ ∃𝑦 𝑥 E 𝑦)
8	5, 7	mpbir 234	. 2 ⊢ 𝑥 ∈ dom E
9	1, 8	mpgbir 1831	1 ⊢ dom E = V

This proof depends on syntax axioms:   = wceq 1569  ∃wex 1811   ∈ wcel 2144  Vcvv 3450   class class class wbr 5102   E cep 5549  dom cdm 5650

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-eprel 5550  df-dm 5660

This theorem is used by: (None)