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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 6193 | . . 3 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴) | |
2 | eqss 3995 | . . 3 ⊢ (◡◡𝐴 = 𝐴 ↔ (◡◡𝐴 ⊆ 𝐴 ∧ 𝐴 ⊆ ◡◡𝐴)) | |
3 | 1, 2 | bitri 275 | . 2 ⊢ (Rel 𝐴 ↔ (◡◡𝐴 ⊆ 𝐴 ∧ 𝐴 ⊆ ◡◡𝐴)) |
4 | cnvcnvss 6198 | . . 3 ⊢ ◡◡𝐴 ⊆ 𝐴 | |
5 | 4 | biantrur 530 | . 2 ⊢ (𝐴 ⊆ ◡◡𝐴 ↔ (◡◡𝐴 ⊆ 𝐴 ∧ 𝐴 ⊆ ◡◡𝐴)) |
6 | ssdif0 4364 | . 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 205 ∧ wa 395 = wceq 1534 ∖ cdif 3944 ⊆ wss 3947 ∅c0 4323 ◡ccnv 5677 Rel wrel 5683 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-opab 5211 df-xp 5684 df-rel 5685 df-cnv 5686 |
This theorem is referenced by: cnvnonrel 43018 |
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