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Theorem 0nelrel 4523
Description: A binary relation does not contain the empty set. (Contributed by AV, 15-Nov-2021.)
Assertion
Ref Expression
0nelrel (Rel 𝑅 → ∅ ∉ 𝑅)

Proof of Theorem 0nelrel
StepHypRef Expression
1 df-rel 4484 . . . 4 (Rel 𝑅𝑅 ⊆ (V × V))
21biimpi 119 . . 3 (Rel 𝑅𝑅 ⊆ (V × V))
3 0nelxp 4505 . . . 4 ¬ ∅ ∈ (V × V)
43a1i 9 . . 3 (Rel 𝑅 → ¬ ∅ ∈ (V × V))
52, 4ssneldd 3050 . 2 (Rel 𝑅 → ¬ ∅ ∈ 𝑅)
6 df-nel 2363 . 2 (∅ ∉ 𝑅 ↔ ¬ ∅ ∈ 𝑅)
75, 6sylibr 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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