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Theorem 0nelrel0 5647
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 5596 . . 3 (Rel 𝑅𝑅 ⊆ (V × V))
21biimpi 215 . 2 (Rel 𝑅𝑅 ⊆ (V × V))
3 0nelxp 5623 . . 3 ¬ ∅ ∈ (V × V)
43a1i 11 . 2 (Rel 𝑅 → ¬ ∅ ∈ (V × V))
52, 4ssneldd 3924 1 (Rel 𝑅 → ¬ ∅ ∈ 𝑅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2106  Vcvv 3432  wss 3887  c0 4256   × cxp 5587  Rel wrel 5594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-opab 5137  df-xp 5595  df-rel 5596
This theorem is referenced by:  0nelrel  5648  reldmtpos  8050  bj-0nelopab  35237  bj-brrelex12ALT  35238
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