Theorem 0nelrel0 5711

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 5658	. . 3 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V))
2	1	biimpi 219	. 2 ⊢ (Rel 𝑅 → 𝑅 ⊆ (V × V))
3		0nelxp 5685	. . 3 ⊢ ¬ ∅ ∈ (V × V)
4	3	a1i 11	. 2 ⊢ (Rel 𝑅 → ¬ ∅ ∈ (V × V))
5	2, 4	ssneldd 3934	1 ⊢ (Rel 𝑅 → ¬ ∅ ∈ 𝑅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∈ wcel 2145  Vcvv 3451   ⊆ wss 3899  ∅c0 4279   × cxp 5649  Rel wrel 5656

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-opab 5168  df-xp 5657  df-rel 5658

This theorem is used by:  0nelrel  5712  nrelv  5777  reldmtpos  8251  bj-0nelopab  37981  bj-brrelex12ALT  37982  tposrescnv  49986  tposres3  49988  tposres  49989  idfurcl  50205  oppfrcllem  50234  2oppf  50239  fucofvalne  50432