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| Mirrors > Home > MPE Home > Th. List > Mathboxes > relnonrel | Structured version Visualization version GIF version | ||
| Description: The non-relation part of a relation is empty. (Contributed by RP, 22-Oct-2020.) | 
| Ref | Expression | 
|---|---|
| relnonrel | ⊢ (Rel 𝐴 ↔ (𝐴 ∖ ◡◡𝐴) = ∅) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | dfrel2 6209 | . . 3 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴) | |
| 2 | eqss 3999 | . . 3 ⊢ (◡◡𝐴 = 𝐴 ↔ (◡◡𝐴 ⊆ 𝐴 ∧ 𝐴 ⊆ ◡◡𝐴)) | |
| 3 | 1, 2 | bitri 275 | . 2 ⊢ (Rel 𝐴 ↔ (◡◡𝐴 ⊆ 𝐴 ∧ 𝐴 ⊆ ◡◡𝐴)) | 
| 4 | cnvcnvss 6214 | . . 3 ⊢ ◡◡𝐴 ⊆ 𝐴 | |
| 5 | 4 | biantrur 530 | . 2 ⊢ (𝐴 ⊆ ◡◡𝐴 ↔ (◡◡𝐴 ⊆ 𝐴 ∧ 𝐴 ⊆ ◡◡𝐴)) | 
| 6 | ssdif0 4366 | . 2 ⊢ (𝐴 ⊆ ◡◡𝐴 ↔ (𝐴 ∖ ◡◡𝐴) = ∅) | |
| 7 | 3, 5, 6 | 3bitr2i 299 | 1 ⊢ (Rel 𝐴 ↔ (𝐴 ∖ ◡◡𝐴) = ∅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∖ cdif 3948 ⊆ wss 3951 ∅c0 4333 ◡ccnv 5684 Rel wrel 5690 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 | 
| This theorem is referenced by: cnvnonrel 43601 | 
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