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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 4605 | . . . 4 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V)) | |
2 | 1 | biimpi 119 | . . 3 ⊢ (Rel 𝑅 → 𝑅 ⊆ (V × V)) |
3 | 0nelxp 4626 | . . . 4 ⊢ ¬ ∅ ∈ (V × V) | |
4 | 3 | a1i 9 | . . 3 ⊢ (Rel 𝑅 → ¬ ∅ ∈ (V × V)) |
5 | 2, 4 | ssneldd 3140 | . 2 ⊢ (Rel 𝑅 → ¬ ∅ ∈ 𝑅) |
6 | df-nel 2430 | . 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 2135 ∉ wnel 2429 Vcvv 2721 ⊆ wss 3111 ∅c0 3404 × cxp 4596 Rel wrel 4603 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-opab 4038 df-xp 4604 df-rel 4605 |
This theorem is referenced by: 0nelfun 5200 |
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