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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 5243 | . . 3 ⊢ Rel ∅ | |
2 | eqrel 5209 | . . 3 ⊢ ((Rel 𝐴 ∧ Rel ∅) → (𝐴 = ∅ ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ ∅))) | |
3 | 1, 2 | mpan2 707 | . 2 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ ∅))) |
4 | eq0 3929 | . . 3 ⊢ (dom 𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom 𝐴) | |
5 | alnex 1706 | . . . . . 6 ⊢ (∀𝑦 ¬ 〈𝑥, 𝑦〉 ∈ 𝐴 ↔ ¬ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴) | |
6 | vex 3203 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
7 | 6 | eldm2 5322 | . . . . . 6 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴) |
8 | 5, 7 | xchbinxr 325 | . . . . 5 ⊢ (∀𝑦 ¬ 〈𝑥, 𝑦〉 ∈ 𝐴 ↔ ¬ 𝑥 ∈ dom 𝐴) |
9 | noel 3919 | . . . . . . 7 ⊢ ¬ 〈𝑥, 𝑦〉 ∈ ∅ | |
10 | 9 | nbn 362 | . . . . . 6 ⊢ (¬ 〈𝑥, 𝑦〉 ∈ 𝐴 ↔ (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ ∅)) |
11 | 10 | albii 1747 | . . . . 5 ⊢ (∀𝑦 ¬ 〈𝑥, 𝑦〉 ∈ 𝐴 ↔ ∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ ∅)) |
12 | 8, 11 | bitr3i 266 | . . . 4 ⊢ (¬ 𝑥 ∈ dom 𝐴 ↔ ∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ ∅)) |
13 | 12 | albii 1747 | . . 3 ⊢ (∀𝑥 ¬ 𝑥 ∈ dom 𝐴 ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ ∅)) |
14 | 4, 13 | bitr2i 265 | . 2 ⊢ (∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ ∅) ↔ dom 𝐴 = ∅) |
15 | 3, 14 | syl6bb 276 | 1 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ dom 𝐴 = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∀wal 1481 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ∅c0 3915 〈cop 4183 dom cdm 5114 Rel wrel 5119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-dm 5124 |
This theorem is referenced by: relrn0 5383 coeq0 5644 fnresdisj 6001 fn0 6011 fresaunres2 6076 funopsn 6413 fsnunfv 6453 frxp 7287 domss2 8119 swrd0 13434 setsres 15901 pmtrsn 17939 gsumval3 18308 00lsp 18981 metn0 22165 wlkn0 26516 eulerpath 27101 funresdm1 29416 dfrdg2 31701 mbfresfi 33456 mapfzcons1 37280 diophrw 37322 eldioph2lem1 37323 eldioph2lem2 37324 sge0cl 40598 |
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