Theorem relnonrel 43628

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 6136	. . 3 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴)
2		eqss 3945	. . 3 ⊢ (◡◡𝐴 = 𝐴 ↔ (◡◡𝐴 ⊆ 𝐴 ∧ 𝐴 ⊆ ◡◡𝐴))
3	1, 2	bitri 275	. 2 ⊢ (Rel 𝐴 ↔ (◡◡𝐴 ⊆ 𝐴 ∧ 𝐴 ⊆ ◡◡𝐴))
4		cnvcnvss 6141	. . 3 ⊢ ◡◡𝐴 ⊆ 𝐴
5	4	biantrur 530	. 2 ⊢ (𝐴 ⊆ ◡◡𝐴 ↔ (◡◡𝐴 ⊆ 𝐴 ∧ 𝐴 ⊆ ◡◡𝐴))
6		ssdif0 4313	. 2 ⊢ (𝐴 ⊆ ◡◡𝐴 ↔ (𝐴 ∖ ◡◡𝐴) = ∅)
7	3, 5, 6	3bitr2i 299	1 ⊢ (Rel 𝐴 ↔ (𝐴 ∖ ◡◡𝐴) = ∅)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541   ∖ cdif 3894   ⊆ wss 3897  ∅c0 4280  ^◡ccnv 5613  Rel wrel 5619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-xp 5620  df-rel 5621  df-cnv 5622

This theorem is referenced by:  cnvnonrel  43629