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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 6188 | . . 3 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴) | |
2 | eqss 3988 | . . 3 ⊢ (◡◡𝐴 = 𝐴 ↔ (◡◡𝐴 ⊆ 𝐴 ∧ 𝐴 ⊆ ◡◡𝐴)) | |
3 | 1, 2 | bitri 274 | . 2 ⊢ (Rel 𝐴 ↔ (◡◡𝐴 ⊆ 𝐴 ∧ 𝐴 ⊆ ◡◡𝐴)) |
4 | cnvcnvss 6193 | . . 3 ⊢ ◡◡𝐴 ⊆ 𝐴 | |
5 | 4 | biantrur 529 | . 2 ⊢ (𝐴 ⊆ ◡◡𝐴 ↔ (◡◡𝐴 ⊆ 𝐴 ∧ 𝐴 ⊆ ◡◡𝐴)) |
6 | ssdif0 4359 | . 2 ⊢ (𝐴 ⊆ ◡◡𝐴 ↔ (𝐴 ∖ ◡◡𝐴) = ∅) | |
7 | 3, 5, 6 | 3bitr2i 298 | 1 ⊢ (Rel 𝐴 ↔ (𝐴 ∖ ◡◡𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1533 ∖ cdif 3936 ⊆ wss 3939 ∅c0 4318 ◡ccnv 5671 Rel wrel 5677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5144 df-opab 5206 df-xp 5678 df-rel 5679 df-cnv 5680 |
This theorem is referenced by: cnvnonrel 43083 |
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