Theorem 0nelrel 5707

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 5706	. 2 ⊢ (Rel 𝑅 → ¬ ∅ ∈ 𝑅)
2		df-nel 3032	. 2 ⊢ (∅ ∉ 𝑅 ↔ ¬ ∅ ∈ 𝑅)
3	1, 2	sylibr 234	1 ⊢ (Rel 𝑅 → ∅ ∉ 𝑅)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2109   ∉ wnel 3031  ∅c0 4304  Rel wrel 5651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5259  ax-nul 5269  ax-pr 5395

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2928  df-nel 3032  df-v 3457  df-dif 3925  df-un 3927  df-ss 3939  df-nul 4305  df-if 4497  df-sn 4598  df-pr 4600  df-op 4604  df-opab 5178  df-xp 5652  df-rel 5653

This theorem is referenced by:  0nelfun  6542