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 3414 | . 2 ⊢ (dom E = V ↔ ∀𝑥 𝑥 ∈ dom E ) | |
2 | el 5259 | . . . 4 ⊢ ∃𝑦 𝑥 ∈ 𝑦 | |
3 | epel 5460 | . . . . 5 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦) | |
4 | 3 | exbii 1855 | . . . 4 ⊢ (∃𝑦 𝑥 E 𝑦 ↔ ∃𝑦 𝑥 ∈ 𝑦) |
5 | 2, 4 | mpbir 234 | . . 3 ⊢ ∃𝑦 𝑥 E 𝑦 |
6 | vex 3409 | . . . 4 ⊢ 𝑥 ∈ V | |
7 | 6 | eldm 5766 | . . 3 ⊢ (𝑥 ∈ dom E ↔ ∃𝑦 𝑥 E 𝑦) |
8 | 5, 7 | mpbir 234 | . 2 ⊢ 𝑥 ∈ dom E |
9 | 1, 8 | mpgbir 1807 | 1 ⊢ dom E = V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∃wex 1787 ∈ wcel 2110 Vcvv 3405 class class class wbr 5050 E cep 5456 dom cdm 5548 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 ax-sep 5189 ax-nul 5196 ax-pow 5255 ax-pr 5319 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2940 df-rab 3067 df-v 3407 df-dif 3866 df-un 3868 df-nul 4235 df-if 4437 df-sn 4539 df-pr 4541 df-op 4545 df-br 5051 df-opab 5113 df-eprel 5457 df-dm 5558 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |