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Theorem 0nelrel0 5690
Description: A binary relation does not contain the empty set. (Contributed by AV, 15-Nov-2021.) (Revised by BJ, 14-Jul-2023.)
Assertion
Ref Expression
0nelrel0 (Rel 𝑅 → ¬ ∅ ∈ 𝑅)

Proof of Theorem 0nelrel0
StepHypRef Expression
1 df-rel 5638 . . 3 (Rel 𝑅𝑅 ⊆ (V × V))
21biimpi 215 . 2 (Rel 𝑅𝑅 ⊆ (V × V))
3 0nelxp 5665 . . 3 ¬ ∅ ∈ (V × V)
43a1i 11 . 2 (Rel 𝑅 → ¬ ∅ ∈ (V × V))
52, 4ssneldd 3945 1 (Rel 𝑅 → ¬ ∅ ∈ 𝑅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2106  Vcvv 3443  wss 3908  c0 4280   × cxp 5629  Rel wrel 5636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2942  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-opab 5166  df-xp 5637  df-rel 5638
This theorem is referenced by:  0nelrel  5691  reldmtpos  8157  bj-0nelopab  35475  bj-brrelex12ALT  35476
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