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Mirrors > Home > ILE Home > Th. List > reldm0 | 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 4711 | . . 3 ⊢ Rel ∅ | |
2 | eqrel 4675 | . . 3 ⊢ ((Rel 𝐴 ∧ Rel ∅) → (𝐴 = ∅ ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ ∅))) | |
3 | 1, 2 | mpan2 422 | . 2 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ ∅))) |
4 | eq0 3412 | . . 3 ⊢ (dom 𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom 𝐴) | |
5 | alnex 1479 | . . . . . 6 ⊢ (∀𝑦 ¬ 〈𝑥, 𝑦〉 ∈ 𝐴 ↔ ¬ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴) | |
6 | vex 2715 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
7 | 6 | eldm2 4784 | . . . . . 6 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴) |
8 | 5, 7 | xchbinxr 673 | . . . . 5 ⊢ (∀𝑦 ¬ 〈𝑥, 𝑦〉 ∈ 𝐴 ↔ ¬ 𝑥 ∈ dom 𝐴) |
9 | noel 3398 | . . . . . . 7 ⊢ ¬ 〈𝑥, 𝑦〉 ∈ ∅ | |
10 | 9 | nbn 689 | . . . . . 6 ⊢ (¬ 〈𝑥, 𝑦〉 ∈ 𝐴 ↔ (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ ∅)) |
11 | 10 | albii 1450 | . . . . 5 ⊢ (∀𝑦 ¬ 〈𝑥, 𝑦〉 ∈ 𝐴 ↔ ∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ ∅)) |
12 | 8, 11 | bitr3i 185 | . . . 4 ⊢ (¬ 𝑥 ∈ dom 𝐴 ↔ ∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ ∅)) |
13 | 12 | albii 1450 | . . 3 ⊢ (∀𝑥 ¬ 𝑥 ∈ dom 𝐴 ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ ∅)) |
14 | 4, 13 | bitr2i 184 | . 2 ⊢ (∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ ∅) ↔ dom 𝐴 = ∅) |
15 | 3, 14 | bitrdi 195 | 1 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ dom 𝐴 = ∅)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 ∀wal 1333 = wceq 1335 ∃wex 1472 ∈ wcel 2128 ∅c0 3394 〈cop 3563 dom cdm 4586 Rel wrel 4591 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-br 3966 df-opab 4026 df-xp 4592 df-rel 4593 df-dm 4596 |
This theorem is referenced by: relrn0 4848 fnresdisj 5280 fn0 5289 fsnunfv 5668 setsresg 12239 metn0 12789 |
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