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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 5809 | . . 3 ⊢ Rel ∅ | |
| 2 | eqrel 5794 | . . 3 ⊢ ((Rel 𝐴 ∧ Rel ∅) → (𝐴 = ∅ ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ ∅))) | |
| 3 | 1, 2 | mpan2 691 | . 2 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ ∅))) |
| 4 | eq0 4350 | . . 3 ⊢ (dom 𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom 𝐴) | |
| 5 | alnex 1781 | . . . . . 6 ⊢ (∀𝑦 ¬ 〈𝑥, 𝑦〉 ∈ 𝐴 ↔ ¬ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴) | |
| 6 | vex 3484 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
| 7 | 6 | eldm2 5912 | . . . . . 6 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴) |
| 8 | 5, 7 | xchbinxr 335 | . . . . 5 ⊢ (∀𝑦 ¬ 〈𝑥, 𝑦〉 ∈ 𝐴 ↔ ¬ 𝑥 ∈ dom 𝐴) |
| 9 | noel 4338 | . . . . . . 7 ⊢ ¬ 〈𝑥, 𝑦〉 ∈ ∅ | |
| 10 | 9 | nbn 372 | . . . . . 6 ⊢ (¬ 〈𝑥, 𝑦〉 ∈ 𝐴 ↔ (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ ∅)) |
| 11 | 10 | albii 1819 | . . . . 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 1819 | . . 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 206 ∀wal 1538 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ∅c0 4333 〈cop 4632 dom cdm 5685 Rel wrel 5690 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-dm 5695 |
| This theorem is referenced by: relrn0 5983 relresdm1 6051 coeq0 6275 snres0 6318 fnresdisj 6688 fn0 6699 fresaunres2 6780 funopsn 7168 fsnunfv 7207 frxp 8151 frxp2 8169 frxp3 8176 domss2 9176 swrd0 14696 setsres 17215 pmtrsn 19537 gsumval3 19925 00lsp 20979 metn0 24370 noetasuplem2 27779 noetainflem2 27783 wlkn0 29639 eulerpath 30260 dfrdg2 35796 mbfresfi 37673 mapfzcons1 42728 diophrw 42770 eldioph2lem1 42771 eldioph2lem2 42772 tfsconcatb0 43357 tfsconcat0i 43358 tfsconcat0b 43359 sge0cl 46396 resinsn 48772 resinsnALT 48773 |
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