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 3450 | . 2 ⊢ (𝐴 ∈ dom 𝑅 → 𝐴 ∈ V) | |
2 | n0 4280 | . . 3 ⊢ ([𝐴]𝑅 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅) | |
3 | ecexr 8503 | . . . 4 ⊢ (𝑥 ∈ [𝐴]𝑅 → 𝐴 ∈ V) | |
4 | 3 | exlimiv 1933 | . . 3 ⊢ (∃𝑥 𝑥 ∈ [𝐴]𝑅 → 𝐴 ∈ V) |
5 | 2, 4 | sylbi 216 | . 2 ⊢ ([𝐴]𝑅 ≠ ∅ → 𝐴 ∈ V) |
6 | vex 3436 | . . . . 5 ⊢ 𝑥 ∈ V | |
7 | elecg 8541 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝐴 ∈ V) → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥)) | |
8 | 6, 7 | mpan 687 | . . . 4 ⊢ (𝐴 ∈ V → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥)) |
9 | 8 | exbidv 1924 | . . 3 ⊢ (𝐴 ∈ V → (∃𝑥 𝑥 ∈ [𝐴]𝑅 ↔ ∃𝑥 𝐴𝑅𝑥)) |
10 | 2 | a1i 11 | . . 3 ⊢ (𝐴 ∈ V → ([𝐴]𝑅 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅)) |
11 | eldmg 5807 | . . 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 380 | 1 ⊢ (𝐴 ∈ dom 𝑅 ↔ [𝐴]𝑅 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∃wex 1782 ∈ wcel 2106 ≠ wne 2943 Vcvv 3432 ∅c0 4256 class class class wbr 5074 dom cdm 5589 [cec 8496 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-br 5075 df-opab 5137 df-xp 5595 df-cnv 5597 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-ec 8500 |
This theorem is referenced by: ereldm 8546 elqsn0 8575 ecelqsdm 8576 eceqoveq 8611 divsfval 17258 sylow1lem5 19207 vitalilem2 24773 vitalilem3 24774 dfdm6 36437 dmecd 36440 n0elqs 36461 |
Copyright terms: Public domain | W3C validator |