Theorem 0nelrel 5311

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		df-rel 5265	. . . 4 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V))
2	1	biimpi 206	. . 3 ⊢ (Rel 𝑅 → 𝑅 ⊆ (V × V))
3		0nelxp 5292	. . . 4 ⊢ ¬ ∅ ∈ (V × V)
4	3	a1i 11	. . 3 ⊢ (Rel 𝑅 → ¬ ∅ ∈ (V × V))
5	2, 4	ssneldd 3739	. 2 ⊢ (Rel 𝑅 → ¬ ∅ ∈ 𝑅)
6		df-nel 3028	. 2 ⊢ (∅ ∉ 𝑅 ↔ ¬ ∅ ∈ 𝑅)
7	5, 6	sylibr 224	1 ⊢ (Rel 𝑅 → ∅ ∉ 𝑅)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2131   ∉ wnel 3027  Vcvv 3332   ⊆ wss 3707  ∅c0 4050   × cxp 5256  Rel wrel 5263

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pr 5047

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-v 3334  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-opab 4857  df-xp 5264  df-rel 5265

This theorem is referenced by:  0nelfun  6059