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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 5498 | . 2 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ dom 𝐴 = ∅)) | |
2 | dm0rn0 5497 | . 2 ⊢ (dom 𝐴 = ∅ ↔ ran 𝐴 = ∅) | |
3 | 1, 2 | syl6bb 276 | 1 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ran 𝐴 = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1632 ∅c0 4058 dom cdm 5266 ran crn 5267 Rel wrel 5271 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pr 5055 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-rab 3059 df-v 3342 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-op 4328 df-br 4805 df-opab 4865 df-xp 5272 df-rel 5273 df-cnv 5274 df-dm 5276 df-rn 5277 |
This theorem is referenced by: cnvsn0 5761 coeq0 5805 foconst 6287 fconst5 6635 edg0iedg0 26148 edg0iedg0OLD 26149 edg0usgr 26344 usgr1v0edg 26348 heicant 33757 |
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