Theorem 0nelrel0 5749

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

Step	Hyp	Ref	Expression
1		df-rel 5696	. . 3 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V))
2	1	biimpi 216	. 2 ⊢ (Rel 𝑅 → 𝑅 ⊆ (V × V))
3		0nelxp 5723	. . 3 ⊢ ¬ ∅ ∈ (V × V)
4	3	a1i 11	. 2 ⊢ (Rel 𝑅 → ¬ ∅ ∈ (V × V))
5	2, 4	ssneldd 3998	1 ⊢ (Rel 𝑅 → ¬ ∅ ∈ 𝑅)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2106  Vcvv 3478   ⊆ wss 3963  ∅c0 4339   × cxp 5687  Rel wrel 5694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-opab 5211  df-xp 5695  df-rel 5696

This theorem is referenced by:  0nelrel  5750  reldmtpos  8258  bj-0nelopab  37049  bj-brrelex12ALT  37050