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Mirrors > Home > MPE Home > Th. List > Mathboxes > domep | Structured version Visualization version GIF version |
Description: The domain of the epsilon relation is the universe. (Contributed by Scott Fenton, 27-Oct-2010.) |
Ref | Expression |
---|---|
domep | ⊢ dom E = V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equid 2059 | . . . 4 ⊢ 𝑥 = 𝑥 | |
2 | el 5083 | . . . . 5 ⊢ ∃𝑦 𝑥 ∈ 𝑦 | |
3 | epel 5271 | . . . . . 6 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦) | |
4 | 3 | exbii 1892 | . . . . 5 ⊢ (∃𝑦 𝑥 E 𝑦 ↔ ∃𝑦 𝑥 ∈ 𝑦) |
5 | 2, 4 | mpbir 223 | . . . 4 ⊢ ∃𝑦 𝑥 E 𝑦 |
6 | 1, 5 | 2th 256 | . . 3 ⊢ (𝑥 = 𝑥 ↔ ∃𝑦 𝑥 E 𝑦) |
7 | 6 | abbii 2908 | . 2 ⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑥 ∣ ∃𝑦 𝑥 E 𝑦} |
8 | df-v 3400 | . 2 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥} | |
9 | df-dm 5367 | . 2 ⊢ dom E = {𝑥 ∣ ∃𝑦 𝑥 E 𝑦} | |
10 | 7, 8, 9 | 3eqtr4ri 2813 | 1 ⊢ dom E = V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1601 ∃wex 1823 {cab 2763 Vcvv 3398 class class class wbr 4888 E cep 5267 dom cdm 5357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-br 4889 df-opab 4951 df-eprel 5268 df-dm 5367 |
This theorem is referenced by: (None) |
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