Theorem 0nelrel 5709

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

Step	Hyp	Ref	Expression
1		0nelrel0 5708	. 2 ⊢ (Rel 𝑅 → ¬ ∅ ∈ 𝑅)
2		df-nel 3063	. 2 ⊢ (∅ ∉ 𝑅 ↔ ¬ ∅ ∈ 𝑅)
3	1, 2	sylibr 236	1 ⊢ (Rel 𝑅 → ∅ ∉ 𝑅)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2143   ∉ wnel 3062  ∅c0 4286  Rel wrel 5653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5247  ax-pr 5391

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-nel 3063  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-opab 5164  df-xp 5654  df-rel 5655

This theorem is referenced by:  0nelfun  6540