Theorem relnonrel 44371

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 6189	. . 3 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴)
2		eqss 3953	. . 3 ⊢ (◡◡𝐴 = 𝐴 ↔ (◡◡𝐴 ⊆ 𝐴 ∧ 𝐴 ⊆ ◡◡𝐴))
3	1, 2	bitri 278	. 2 ⊢ (Rel 𝐴 ↔ (◡◡𝐴 ⊆ 𝐴 ∧ 𝐴 ⊆ ◡◡𝐴))
4		cnvcnvss 6194	. . 3 ⊢ ◡◡𝐴 ⊆ 𝐴
5	4	biantrur 540	. 2 ⊢ (𝐴 ⊆ ◡◡𝐴 ↔ (◡◡𝐴 ⊆ 𝐴 ∧ 𝐴 ⊆ ◡◡𝐴))
6		ssdif0 4321	. 2 ⊢ (𝐴 ⊆ ◡◡𝐴 ↔ (𝐴 ∖ ◡◡𝐴) = ∅)
7	3, 5, 6	3bitr2i 302	1 ⊢ (Rel 𝐴 ↔ (𝐴 ∖ ◡◡𝐴) = ∅)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∖ cdif 3903   ⊆ wss 3906  ∅c0 4286  ^◡ccnv 5662  Rel wrel 5668

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-xp 5669  df-rel 5670  df-cnv 5671  df-res 5675

This theorem is used by:  cnvnonrel  44372