Theorem relnonrel 44160

Description: The non-relation part of a relation is empty. (Contributed by RP, 22-Oct-2020.)

Assertion

Ref	Expression
relnonrel	⊢ (Rel 𝐴 ↔ (𝐴 ∖ ◡◡𝐴) = ∅)

Proof of Theorem relnonrel

Step	Hyp	Ref	Expression
1		dfrel2 6175	. . 3 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴)
2		eqss 3951	. . 3 ⊢ (◡◡𝐴 = 𝐴 ↔ (◡◡𝐴 ⊆ 𝐴 ∧ 𝐴 ⊆ ◡◡𝐴))
3	1, 2	bitri 277	. 2 ⊢ (Rel 𝐴 ↔ (◡◡𝐴 ⊆ 𝐴 ∧ 𝐴 ⊆ ◡◡𝐴))
4		cnvcnvss 6180	. . 3 ⊢ ◡◡𝐴 ⊆ 𝐴
5	4	biantrur 538	. 2 ⊢ (𝐴 ⊆ ◡◡𝐴 ↔ (◡◡𝐴 ⊆ 𝐴 ∧ 𝐴 ⊆ ◡◡𝐴))
6		ssdif0 4319	. 2 ⊢ (𝐴 ⊆ ◡◡𝐴 ↔ (𝐴 ∖ ◡◡𝐴) = ∅)
7	3, 5, 6	3bitr2i 301	1 ⊢ (Rel 𝐴 ↔ (𝐴 ∖ ◡◡𝐴) = ∅)

Syntax hints:   ↔ wb 208   ∧ wa 399   = wceq 1560   ∖ cdif 3901   ⊆ wss 3904  ∅c0 4285  ^◡ccnv 5646  Rel wrel 5652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5653  df-rel 5654  df-cnv 5655  df-res 5659

This theorem is referenced by:  cnvnonrel  44161