ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  0nelrel GIF version

Theorem 0nelrel 4687
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 4648 . . . 4 (Rel 𝑅𝑅 ⊆ (V × V))
21biimpi 120 . . 3 (Rel 𝑅𝑅 ⊆ (V × V))
3 0nelxp 4669 . . . 4 ¬ ∅ ∈ (V × V)
43a1i 9 . . 3 (Rel 𝑅 → ¬ ∅ ∈ (V × V))
52, 4ssneldd 3173 . 2 (Rel 𝑅 → ¬ ∅ ∈ 𝑅)
6 df-nel 2456 . 2 (∅ ∉ 𝑅 ↔ ¬ ∅ ∈ 𝑅)
75, 6sylibr 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
  Copyright terms: Public domain W3C validator