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Theorem 0nelrel0 5583
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 5532 . . 3 (Rel 𝑅𝑅 ⊆ (V × V))
21biimpi 219 . 2 (Rel 𝑅𝑅 ⊆ (V × V))
3 0nelxp 5559 . . 3 ¬ ∅ ∈ (V × V)
43a1i 11 . 2 (Rel 𝑅 → ¬ ∅ ∈ (V × V))
52, 4ssneldd 3880 1 (Rel 𝑅 → ¬ ∅ ∈ 𝑅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2114  Vcvv 3398  wss 3843  c0 4211   × cxp 5523  Rel wrel 5530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ne 2935  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-opab 5093  df-xp 5531  df-rel 5532
This theorem is referenced by:  0nelrel  5584  reldmtpos  7929  bj-0nelopab  34859  bj-brrelex12ALT  34860
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