Theorem 0nelrel0 5605

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 5555	. . 3 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V))
2	1	biimpi 218	. 2 ⊢ (Rel 𝑅 → 𝑅 ⊆ (V × V))
3		0nelxp 5582	. . 3 ⊢ ¬ ∅ ∈ (V × V)
4	3	a1i 11	. 2 ⊢ (Rel 𝑅 → ¬ ∅ ∈ (V × V))
5	2, 4	ssneldd 3963	1 ⊢ (Rel 𝑅 → ¬ ∅ ∈ 𝑅)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2113  Vcvv 3491   ⊆ wss 3929  ∅c0 4284   × cxp 5546  Rel wrel 5553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-sep 5196  ax-nul 5203  ax-pr 5323

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ne 3016  df-v 3493  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-nul 4285  df-if 4461  df-sn 4561  df-pr 4563  df-op 4567  df-opab 5122  df-xp 5554  df-rel 5555

This theorem is referenced by:  0nelrel  5606  reldmtpos  7893  bj-0nelopab  34380  bj-brrelex12ALT  34381