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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 4648 | . . . 4 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V)) | |
2 | 1 | biimpi 120 | . . 3 ⊢ (Rel 𝑅 → 𝑅 ⊆ (V × V)) |
3 | 0nelxp 4669 | . . . 4 ⊢ ¬ ∅ ∈ (V × V) | |
4 | 3 | a1i 9 | . . 3 ⊢ (Rel 𝑅 → ¬ ∅ ∈ (V × V)) |
5 | 2, 4 | ssneldd 3173 | . 2 ⊢ (Rel 𝑅 → ¬ ∅ ∈ 𝑅) |
6 | df-nel 2456 | . 2 ⊢ (∅ ∉ 𝑅 ↔ ¬ ∅ ∈ 𝑅) | |
7 | 5, 6 | sylibr 134 | 1 ⊢ (Rel 𝑅 → ∅ ∉ 𝑅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2160 ∉ wnel 2455 Vcvv 2752 ⊆ wss 3144 ∅c0 3437 × cxp 4639 Rel wrel 4646 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-opab 4080 df-xp 4647 df-rel 4648 |
This theorem is referenced by: 0nelfun 5249 |
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