Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > relrn0 | Structured version Visualization version GIF version |
Description: A relation is empty iff its range is empty. (Contributed by NM, 15-Sep-2004.) |
Ref | Expression |
---|---|
relrn0 | ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ran 𝐴 = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reldm0 5791 | . 2 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ dom 𝐴 = ∅)) | |
2 | dm0rn0 5788 | . 2 ⊢ (dom 𝐴 = ∅ ↔ ran 𝐴 = ∅) | |
3 | 1, 2 | syl6bb 288 | 1 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ran 𝐴 = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 ∅c0 4288 dom cdm 5548 ran crn 5549 Rel wrel 5553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-cnv 5556 df-dm 5558 df-rn 5559 |
This theorem is referenced by: cnvsn0 6060 coeq0 6101 foconst 6596 fconst5 6960 edg0iedg0 26767 edg0usgr 26962 usgr1v0edg 26966 tocyccntz 30713 heicant 34808 |
Copyright terms: Public domain | W3C validator |