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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 3480 | . 2 ⊢ (𝐴 ∈ dom 𝑅 → 𝐴 ∈ V) | |
2 | n0 4346 | . . 3 ⊢ ([𝐴]𝑅 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅) | |
3 | ecexr 8730 | . . . 4 ⊢ (𝑥 ∈ [𝐴]𝑅 → 𝐴 ∈ V) | |
4 | 3 | exlimiv 1925 | . . 3 ⊢ (∃𝑥 𝑥 ∈ [𝐴]𝑅 → 𝐴 ∈ V) |
5 | 2, 4 | sylbi 216 | . 2 ⊢ ([𝐴]𝑅 ≠ ∅ → 𝐴 ∈ V) |
6 | vex 3465 | . . . . 5 ⊢ 𝑥 ∈ V | |
7 | elecg 8768 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝐴 ∈ V) → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥)) | |
8 | 6, 7 | mpan 688 | . . . 4 ⊢ (𝐴 ∈ V → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥)) |
9 | 8 | exbidv 1916 | . . 3 ⊢ (𝐴 ∈ V → (∃𝑥 𝑥 ∈ [𝐴]𝑅 ↔ ∃𝑥 𝐴𝑅𝑥)) |
10 | 2 | a1i 11 | . . 3 ⊢ (𝐴 ∈ V → ([𝐴]𝑅 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅)) |
11 | eldmg 5901 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥)) | |
12 | 9, 10, 11 | 3bitr4rd 311 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ dom 𝑅 ↔ [𝐴]𝑅 ≠ ∅)) |
13 | 1, 5, 12 | pm5.21nii 377 | 1 ⊢ (𝐴 ∈ dom 𝑅 ↔ [𝐴]𝑅 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∃wex 1773 ∈ wcel 2098 ≠ wne 2929 Vcvv 3461 ∅c0 4322 class class class wbr 5149 dom cdm 5678 [cec 8723 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5150 df-opab 5212 df-xp 5684 df-cnv 5686 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ec 8727 |
This theorem is referenced by: ereldm 8774 elqsn0 8805 ecelqsdm 8806 eceqoveq 8841 divsfval 17532 sylow1lem5 19569 vitalilem2 25582 vitalilem3 25583 dfdm6 37900 dmecd 37903 n0elqs 37925 |
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