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| 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 3490 | . 2 ⊢ (dom E = V ↔ ∀𝑥 𝑥 ∈ dom E ) | |
| 2 | el 5442 | . . . 4 ⊢ ∃𝑦 𝑥 ∈ 𝑦 | |
| 3 | epel 5587 | . . . . 5 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦) | |
| 4 | 3 | exbii 1848 | . . . 4 ⊢ (∃𝑦 𝑥 E 𝑦 ↔ ∃𝑦 𝑥 ∈ 𝑦) |
| 5 | 2, 4 | mpbir 231 | . . 3 ⊢ ∃𝑦 𝑥 E 𝑦 |
| 6 | vex 3484 | . . . 4 ⊢ 𝑥 ∈ V | |
| 7 | 6 | eldm 5911 | . . 3 ⊢ (𝑥 ∈ dom E ↔ ∃𝑦 𝑥 E 𝑦) |
| 8 | 5, 7 | mpbir 231 | . 2 ⊢ 𝑥 ∈ dom E |
| 9 | 1, 8 | mpgbir 1799 | 1 ⊢ dom E = V |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∃wex 1779 ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 E cep 5583 dom cdm 5685 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-eprel 5584 df-dm 5695 |
| This theorem is referenced by: (None) |
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