Theorem 0nelrel0 5715

Description: A binary relation does not contain the empty set. (Contributed by AV, 15-Nov-2021.) (Revised by BJ, 14-Jul-2023.)

Assertion

Ref	Expression
0nelrel0	⊢ (Rel 𝑅 → ¬ ∅ ∈ 𝑅)

Proof of Theorem 0nelrel0

Step	Hyp	Ref	Expression
1		df-rel 5662	. . 3 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V))
2	1	biimpi 219	. 2 ⊢ (Rel 𝑅 → 𝑅 ⊆ (V × V))
3		0nelxp 5689	. . 3 ⊢ ¬ ∅ ∈ (V × V)
4	3	a1i 11	. 2 ⊢ (Rel 𝑅 → ¬ ∅ ∈ (V × V))
5	2, 4	ssneldd 3916	1 ⊢ (Rel 𝑅 → ¬ ∅ ∈ 𝑅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∈ wcel 2146  Vcvv 3454   ⊆ wss 3900  ∅c0 4281   × cxp 5653  Rel wrel 5660

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-opab 5171  df-xp 5661  df-rel 5662

This theorem is used by:  0nelrel  5716  nrelv  5782  reldmtpos  8265  bj-0nelopab  38123  bj-brrelex12ALT  38124  tposrescnv  50134  tposres3  50136  tposres  50137  idfurcl  50356  oppfrcllem  50385  2oppf  50390  fucofvalne  50589