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Mirrors > Home > ILE Home > Th. List > 0nelrel | GIF version |
Description: A binary relation does not contain the empty set. (Contributed by AV, 15-Nov-2021.) |
Ref | Expression |
---|---|
0nelrel | ⊢ (Rel 𝑅 → ∅ ∉ 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rel 4484 | . . . 4 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V)) | |
2 | 1 | biimpi 119 | . . 3 ⊢ (Rel 𝑅 → 𝑅 ⊆ (V × V)) |
3 | 0nelxp 4505 | . . . 4 ⊢ ¬ ∅ ∈ (V × V) | |
4 | 3 | a1i 9 | . . 3 ⊢ (Rel 𝑅 → ¬ ∅ ∈ (V × V)) |
5 | 2, 4 | ssneldd 3050 | . 2 ⊢ (Rel 𝑅 → ¬ ∅ ∈ 𝑅) |
6 | df-nel 2363 | . 2 ⊢ (∅ ∉ 𝑅 ↔ ¬ ∅ ∈ 𝑅) | |
7 | 5, 6 | sylibr 133 | 1 ⊢ (Rel 𝑅 → ∅ ∉ 𝑅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 1448 ∉ wnel 2362 Vcvv 2641 ⊆ wss 3021 ∅c0 3310 × cxp 4475 Rel wrel 4482 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-nel 2363 df-v 2643 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-opab 3930 df-xp 4483 df-rel 4484 |
This theorem is referenced by: 0nelfun 5077 |
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