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Mirrors > Home > MPE Home > Th. List > reldm0 | Structured version Visualization version GIF version |
Description: A relation is empty iff its domain is empty. (Contributed by NM, 15-Sep-2004.) |
Ref | Expression |
---|---|
reldm0 | ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ dom 𝐴 = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rel0 5470 | . . 3 ⊢ Rel ∅ | |
2 | eqrel 5456 | . . 3 ⊢ ((Rel 𝐴 ∧ Rel ∅) → (𝐴 = ∅ ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ ∅))) | |
3 | 1, 2 | mpan2 681 | . 2 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ ∅))) |
4 | eq0 4157 | . . 3 ⊢ (dom 𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom 𝐴) | |
5 | alnex 1825 | . . . . . 6 ⊢ (∀𝑦 ¬ 〈𝑥, 𝑦〉 ∈ 𝐴 ↔ ¬ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴) | |
6 | vex 3401 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
7 | 6 | eldm2 5567 | . . . . . 6 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴) |
8 | 5, 7 | xchbinxr 327 | . . . . 5 ⊢ (∀𝑦 ¬ 〈𝑥, 𝑦〉 ∈ 𝐴 ↔ ¬ 𝑥 ∈ dom 𝐴) |
9 | noel 4146 | . . . . . . 7 ⊢ ¬ 〈𝑥, 𝑦〉 ∈ ∅ | |
10 | 9 | nbn 364 | . . . . . 6 ⊢ (¬ 〈𝑥, 𝑦〉 ∈ 𝐴 ↔ (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ ∅)) |
11 | 10 | albii 1863 | . . . . 5 ⊢ (∀𝑦 ¬ 〈𝑥, 𝑦〉 ∈ 𝐴 ↔ ∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ ∅)) |
12 | 8, 11 | bitr3i 269 | . . . 4 ⊢ (¬ 𝑥 ∈ dom 𝐴 ↔ ∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ ∅)) |
13 | 12 | albii 1863 | . . 3 ⊢ (∀𝑥 ¬ 𝑥 ∈ dom 𝐴 ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ ∅)) |
14 | 4, 13 | bitr2i 268 | . 2 ⊢ (∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ ∅) ↔ dom 𝐴 = ∅) |
15 | 3, 14 | syl6bb 279 | 1 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ dom 𝐴 = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∀wal 1599 = wceq 1601 ∃wex 1823 ∈ wcel 2107 ∅c0 4141 〈cop 4404 dom cdm 5355 Rel wrel 5360 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-br 4887 df-opab 4949 df-xp 5361 df-rel 5362 df-dm 5365 |
This theorem is referenced by: relrn0 5629 coeq0 5898 fnresdisj 6247 fn0 6257 fresaunres2 6326 funopsn 6679 fsnunfv 6724 frxp 7568 domss2 8407 swrd0 13753 setsres 16297 pmtrsn 18323 gsumval3 18694 00lsp 19376 metn0 22573 wlkn0 26968 eulerpath 27645 funresdm1 29979 dfrdg2 32289 mbfresfi 34081 mapfzcons1 38240 diophrw 38282 eldioph2lem1 38283 eldioph2lem2 38284 sge0cl 41522 |
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