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Theorem 0nelrel 5693
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 0nelrel0 5692 . 2 (Rel 𝑅 → ¬ ∅ ∈ 𝑅)
2 df-nel 3050 . 2 (∅ ∉ 𝑅 ↔ ¬ ∅ ∈ 𝑅)
31, 2sylibr 233 1 (Rel 𝑅 → ∅ ∉ 𝑅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2106  wnel 3049  c0 4282  Rel wrel 5638
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 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
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 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-nel 3050  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-opab 5168  df-xp 5639  df-rel 5640
This theorem is referenced by:  0nelfun  6519
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