![]() |
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 3464 | . 2 ⊢ (𝐴 ∈ dom 𝑅 → 𝐴 ∈ V) | |
2 | n0 4307 | . . 3 ⊢ ([𝐴]𝑅 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅) | |
3 | ecexr 8654 | . . . 4 ⊢ (𝑥 ∈ [𝐴]𝑅 → 𝐴 ∈ V) | |
4 | 3 | exlimiv 1934 | . . 3 ⊢ (∃𝑥 𝑥 ∈ [𝐴]𝑅 → 𝐴 ∈ V) |
5 | 2, 4 | sylbi 216 | . 2 ⊢ ([𝐴]𝑅 ≠ ∅ → 𝐴 ∈ V) |
6 | vex 3450 | . . . . 5 ⊢ 𝑥 ∈ V | |
7 | elecg 8692 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝐴 ∈ V) → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥)) | |
8 | 6, 7 | mpan 689 | . . . 4 ⊢ (𝐴 ∈ V → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥)) |
9 | 8 | exbidv 1925 | . . 3 ⊢ (𝐴 ∈ V → (∃𝑥 𝑥 ∈ [𝐴]𝑅 ↔ ∃𝑥 𝐴𝑅𝑥)) |
10 | 2 | a1i 11 | . . 3 ⊢ (𝐴 ∈ V → ([𝐴]𝑅 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅)) |
11 | eldmg 5855 | . . 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 2107 ≠ wne 2944 Vcvv 3446 ∅c0 4283 class class class wbr 5106 dom cdm 5634 [cec 8647 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-xp 5640 df-cnv 5642 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-ec 8651 |
This theorem is referenced by: ereldm 8697 elqsn0 8726 ecelqsdm 8727 eceqoveq 8762 divsfval 17430 sylow1lem5 19385 vitalilem2 24976 vitalilem3 24977 dfdm6 36765 dmecd 36768 n0elqs 36790 |
Copyright terms: Public domain | W3C validator |