![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ecdmn0 | Structured version Visualization version GIF version |
Description: A representative of a nonempty equivalence class belongs to the domain of the equivalence relation. (Contributed by NM, 15-Feb-1996.) (Revised by Mario Carneiro, 9-Jul-2014.) |
Ref | Expression |
---|---|
ecdmn0 | ⊢ (𝐴 ∈ dom 𝑅 ↔ [𝐴]𝑅 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3499 | . 2 ⊢ (𝐴 ∈ dom 𝑅 → 𝐴 ∈ V) | |
2 | n0 4359 | . . 3 ⊢ ([𝐴]𝑅 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅) | |
3 | ecexr 8749 | . . . 4 ⊢ (𝑥 ∈ [𝐴]𝑅 → 𝐴 ∈ V) | |
4 | 3 | exlimiv 1928 | . . 3 ⊢ (∃𝑥 𝑥 ∈ [𝐴]𝑅 → 𝐴 ∈ V) |
5 | 2, 4 | sylbi 217 | . 2 ⊢ ([𝐴]𝑅 ≠ ∅ → 𝐴 ∈ V) |
6 | vex 3482 | . . . . 5 ⊢ 𝑥 ∈ V | |
7 | elecg 8788 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝐴 ∈ V) → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥)) | |
8 | 6, 7 | mpan 690 | . . . 4 ⊢ (𝐴 ∈ V → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥)) |
9 | 8 | exbidv 1919 | . . 3 ⊢ (𝐴 ∈ V → (∃𝑥 𝑥 ∈ [𝐴]𝑅 ↔ ∃𝑥 𝐴𝑅𝑥)) |
10 | 2 | a1i 11 | . . 3 ⊢ (𝐴 ∈ V → ([𝐴]𝑅 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅)) |
11 | eldmg 5912 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥)) | |
12 | 9, 10, 11 | 3bitr4rd 312 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ dom 𝑅 ↔ [𝐴]𝑅 ≠ ∅)) |
13 | 1, 5, 12 | pm5.21nii 378 | 1 ⊢ (𝐴 ∈ dom 𝑅 ↔ [𝐴]𝑅 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∃wex 1776 ∈ wcel 2106 ≠ wne 2938 Vcvv 3478 ∅c0 4339 class class class wbr 5148 dom cdm 5689 [cec 8742 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ec 8746 |
This theorem is referenced by: ereldm 8794 elqsn0 8825 ecelqsdm 8826 eceqoveq 8861 divsfval 17594 sylow1lem5 19635 vitalilem2 25658 vitalilem3 25659 dfdm6 38283 dmecd 38286 n0elqs 38308 |
Copyright terms: Public domain | W3C validator |