Theorem relnonrel 36912

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 5502	. . 3 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴)
2		eqss 3583	. . 3 ⊢ (◡◡𝐴 = 𝐴 ↔ (◡◡𝐴 ⊆ 𝐴 ∧ 𝐴 ⊆ ◡◡𝐴))
3	1, 2	bitri 263	. 2 ⊢ (Rel 𝐴 ↔ (◡◡𝐴 ⊆ 𝐴 ∧ 𝐴 ⊆ ◡◡𝐴))
4		cnvcnvss 5507	. . 3 ⊢ ◡◡𝐴 ⊆ 𝐴
5	4	biantrur 526	. 2 ⊢ (𝐴 ⊆ ◡◡𝐴 ↔ (◡◡𝐴 ⊆ 𝐴 ∧ 𝐴 ⊆ ◡◡𝐴))
6		ssdif0 3896	. 2 ⊢ (𝐴 ⊆ ◡◡𝐴 ↔ (𝐴 ∖ ◡◡𝐴) = ∅)
7	3, 5, 6	3bitr2i 287	1 ⊢ (Rel 𝐴 ↔ (𝐴 ∖ ◡◡𝐴) = ∅)

Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  ^◡ccnv 5037  Rel wrel 5043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046

This theorem is referenced by:  cnvnonrel  36913