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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 5756 | . . 3 ⊢ Rel ∅ | |
2 | eqrel 5741 | . . 3 ⊢ ((Rel 𝐴 ∧ Rel ∅) → (𝐴 = ∅ ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅))) | |
3 | 1, 2 | mpan2 690 | . 2 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅))) |
4 | eq0 4304 | . . 3 ⊢ (dom 𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom 𝐴) | |
5 | alnex 1784 | . . . . . 6 ⊢ (∀𝑦 ¬ ⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ¬ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴) | |
6 | vex 3448 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
7 | 6 | eldm2 5858 | . . . . . 6 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴) |
8 | 5, 7 | xchbinxr 335 | . . . . 5 ⊢ (∀𝑦 ¬ ⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ¬ 𝑥 ∈ dom 𝐴) |
9 | noel 4291 | . . . . . . 7 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅ | |
10 | 9 | nbn 373 | . . . . . 6 ⊢ (¬ ⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)) |
11 | 10 | albii 1822 | . . . . 5 ⊢ (∀𝑦 ¬ ⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)) |
12 | 8, 11 | bitr3i 277 | . . . 4 ⊢ (¬ 𝑥 ∈ dom 𝐴 ↔ ∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)) |
13 | 12 | albii 1822 | . . 3 ⊢ (∀𝑥 ¬ 𝑥 ∈ dom 𝐴 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)) |
14 | 4, 13 | bitr2i 276 | . 2 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅) ↔ dom 𝐴 = ∅) |
15 | 3, 14 | bitrdi 287 | 1 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ dom 𝐴 = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∀wal 1540 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ∅c0 4283 ⟨cop 4593 dom cdm 5634 Rel wrel 5639 |
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-ext 2704 |
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-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3407 df-v 3446 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-rel 5641 df-dm 5644 |
This theorem is referenced by: relrn0 5925 coeq0 6208 snres0 6251 fnresdisj 6622 fn0 6633 fresaunres2 6715 funopsn 7095 fsnunfv 7134 frxp 8059 frxp2 8077 frxp3 8084 domss2 9083 swrd0 14552 setsres 17055 pmtrsn 19306 gsumval3 19689 00lsp 20457 metn0 23729 noetasuplem2 27098 noetainflem2 27102 wlkn0 28611 eulerpath 29227 funresdm1 31569 dfrdg2 34426 mbfresfi 36170 mapfzcons1 41083 diophrw 41125 eldioph2lem1 41126 eldioph2lem2 41127 sge0cl 44708 |
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