Theorem 0nelrel 5723

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 5722	. 2 ⊢ (Rel 𝑅 → ¬ ∅ ∈ 𝑅)
2		df-nel 3071	. 2 ⊢ (∅ ∉ 𝑅 ↔ ¬ ∅ ∈ 𝑅)
3	1, 2	sylibr 237	1 ⊢ (Rel 𝑅 → ∅ ∉ 𝑅)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2149   ∉ wnel 3070  ∅c0 4294  Rel wrel 5667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-nel 3071  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-opab 5178  df-xp 5668  df-rel 5669

This theorem is referenced by:  0nelfun  6555