![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > dmep | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
dmep | ⊢ dom E = V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqv 3484 | . 2 ⊢ (dom E = V ↔ ∀𝑥 𝑥 ∈ dom E ) | |
2 | el 5438 | . . . 4 ⊢ ∃𝑦 𝑥 ∈ 𝑦 | |
3 | epel 5584 | . . . . 5 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦) | |
4 | 3 | exbii 1851 | . . . 4 ⊢ (∃𝑦 𝑥 E 𝑦 ↔ ∃𝑦 𝑥 ∈ 𝑦) |
5 | 2, 4 | mpbir 230 | . . 3 ⊢ ∃𝑦 𝑥 E 𝑦 |
6 | vex 3479 | . . . 4 ⊢ 𝑥 ∈ V | |
7 | 6 | eldm 5901 | . . 3 ⊢ (𝑥 ∈ dom E ↔ ∃𝑦 𝑥 E 𝑦) |
8 | 5, 7 | mpbir 230 | . 2 ⊢ 𝑥 ∈ dom E |
9 | 1, 8 | mpgbir 1802 | 1 ⊢ dom E = V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∃wex 1782 ∈ wcel 2107 Vcvv 3475 class class class wbr 5149 E cep 5580 dom cdm 5677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-eprel 5581 df-dm 5687 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |