Theorem 0nelrel 5680

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 5679	. 2 ⊢ (Rel 𝑅 → ¬ ∅ ∈ 𝑅)
2		df-nel 3039	. 2 ⊢ (∅ ∉ 𝑅 ↔ ¬ ∅ ∈ 𝑅)
3	1, 2	sylibr 235	1 ⊢ (Rel 𝑅 → ∅ ∉ 𝑅)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2119   ∉ wnel 3038  ∅c0 4262  Rel wrel 5624

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5219  ax-pr 5363

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-nel 3039  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-sn 4557  df-pr 4559  df-op 4563  df-opab 5136  df-xp 5625  df-rel 5626

This theorem is referenced by:  0nelfun  6504