Theorem relnonrel 44031

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 6140	. . 3 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴)
2		eqss 3930	. . 3 ⊢ (◡◡𝐴 = 𝐴 ↔ (◡◡𝐴 ⊆ 𝐴 ∧ 𝐴 ⊆ ◡◡𝐴))
3	1, 2	bitri 276	. 2 ⊢ (Rel 𝐴 ↔ (◡◡𝐴 ⊆ 𝐴 ∧ 𝐴 ⊆ ◡◡𝐴))
4		cnvcnvss 6145	. . 3 ⊢ ◡◡𝐴 ⊆ 𝐴
5	4	biantrur 535	. 2 ⊢ (𝐴 ⊆ ◡◡𝐴 ↔ (◡◡𝐴 ⊆ 𝐴 ∧ 𝐴 ⊆ ◡◡𝐴))
6		ssdif0 4294	. 2 ⊢ (𝐴 ⊆ ◡◡𝐴 ↔ (𝐴 ∖ ◡◡𝐴) = ∅)
7	3, 5, 6	3bitr2i 300	1 ⊢ (Rel 𝐴 ↔ (𝐴 ∖ ◡◡𝐴) = ∅)

Syntax hints:   ↔ wb 207   ∧ wa 396   = wceq 1547   ∖ cdif 3880   ⊆ wss 3883  ∅c0 4261  ^◡ccnv 5617  Rel wrel 5623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-xp 5624  df-rel 5625  df-cnv 5626

This theorem is referenced by:  cnvnonrel  44032