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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 5636 | . . 3 ⊢ Rel ∅ | |
2 | eqrel 5622 | . . 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 4258 | . . 3 ⊢ (dom 𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom 𝐴) | |
5 | alnex 1783 | . . . . . 6 ⊢ (∀𝑦 ¬ 〈𝑥, 𝑦〉 ∈ 𝐴 ↔ ¬ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴) | |
6 | vex 3444 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
7 | 6 | eldm2 5734 | . . . . . 6 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴) |
8 | 5, 7 | xchbinxr 338 | . . . . 5 ⊢ (∀𝑦 ¬ 〈𝑥, 𝑦〉 ∈ 𝐴 ↔ ¬ 𝑥 ∈ dom 𝐴) |
9 | noel 4247 | . . . . . . 7 ⊢ ¬ 〈𝑥, 𝑦〉 ∈ ∅ | |
10 | 9 | nbn 376 | . . . . . 6 ⊢ (¬ 〈𝑥, 𝑦〉 ∈ 𝐴 ↔ (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ ∅)) |
11 | 10 | albii 1821 | . . . . 5 ⊢ (∀𝑦 ¬ 〈𝑥, 𝑦〉 ∈ 𝐴 ↔ ∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ ∅)) |
12 | 8, 11 | bitr3i 280 | . . . 4 ⊢ (¬ 𝑥 ∈ dom 𝐴 ↔ ∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ ∅)) |
13 | 12 | albii 1821 | . . 3 ⊢ (∀𝑥 ¬ 𝑥 ∈ dom 𝐴 ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ ∅)) |
14 | 4, 13 | bitr2i 279 | . 2 ⊢ (∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ ∅) ↔ dom 𝐴 = ∅) |
15 | 3, 14 | syl6bb 290 | 1 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ dom 𝐴 = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∀wal 1536 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ∅c0 4243 〈cop 4531 dom cdm 5519 Rel wrel 5524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-dm 5529 |
This theorem is referenced by: relrn0 5805 coeq0 6075 fnresdisj 6439 fn0 6451 fresaunres2 6524 funopsn 6887 fsnunfv 6926 frxp 7803 domss2 8660 swrd0 14011 setsres 16517 pmtrsn 18639 gsumval3 19020 00lsp 19746 metn0 22967 wlkn0 27410 eulerpath 28026 funresdm1 30368 dfrdg2 33153 mbfresfi 35103 mapfzcons1 39658 diophrw 39700 eldioph2lem1 39701 eldioph2lem2 39702 sge0cl 43020 |
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