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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 1939 | . . . 4 ⊢ 𝑥 = 𝑥 | |
2 | el 4847 | . . . . 5 ⊢ ∃𝑦 𝑥 ∈ 𝑦 | |
3 | epel 5032 | . . . . . 6 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦) | |
4 | 3 | exbii 1774 | . . . . 5 ⊢ (∃𝑦 𝑥 E 𝑦 ↔ ∃𝑦 𝑥 ∈ 𝑦) |
5 | 2, 4 | mpbir 221 | . . . 4 ⊢ ∃𝑦 𝑥 E 𝑦 |
6 | 1, 5 | 2th 254 | . . 3 ⊢ (𝑥 = 𝑥 ↔ ∃𝑦 𝑥 E 𝑦) |
7 | 6 | abbii 2739 | . 2 ⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑥 ∣ ∃𝑦 𝑥 E 𝑦} |
8 | df-v 3202 | . 2 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥} | |
9 | df-dm 5124 | . 2 ⊢ dom E = {𝑥 ∣ ∃𝑦 𝑥 E 𝑦} | |
10 | 7, 8, 9 | 3eqtr4ri 2655 | 1 ⊢ dom E = V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∃wex 1704 {cab 2608 Vcvv 3200 class class class wbr 4653 E cep 5028 dom cdm 5114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-eprel 5029 df-dm 5124 |
This theorem is referenced by: (None) |
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