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Mirrors > Home > ILE Home > Th. List > relrn0 | 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 4827 | . 2 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ dom 𝐴 = ∅)) | |
2 | dm0rn0 4826 | . 2 ⊢ (dom 𝐴 = ∅ ↔ ran 𝐴 = ∅) | |
3 | 1, 2 | bitrdi 195 | 1 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ran 𝐴 = ∅)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1348 ∅c0 3414 dom cdm 4609 ran crn 4610 Rel wrel 4614 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-br 3988 df-opab 4049 df-xp 4615 df-rel 4616 df-cnv 4617 df-dm 4619 df-rn 4620 |
This theorem is referenced by: cnvsn0 5077 |
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