Theorem 0nelrel0 5760

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 5707	. . 3 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V))
2	1	biimpi 216	. 2 ⊢ (Rel 𝑅 → 𝑅 ⊆ (V × V))
3		0nelxp 5734	. . 3 ⊢ ¬ ∅ ∈ (V × V)
4	3	a1i 11	. 2 ⊢ (Rel 𝑅 → ¬ ∅ ∈ (V × V))
5	2, 4	ssneldd 4011	1 ⊢ (Rel 𝑅 → ¬ ∅ ∈ 𝑅)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2108  Vcvv 3488   ⊆ wss 3976  ∅c0 4352   × cxp 5698  Rel wrel 5705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-opab 5229  df-xp 5706  df-rel 5707

This theorem is referenced by:  0nelrel  5761  reldmtpos  8275  bj-0nelopab  37032  bj-brrelex12ALT  37033