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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 5824 | . . 3 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴) | |
2 | eqss 3842 | . . 3 ⊢ (◡◡𝐴 = 𝐴 ↔ (◡◡𝐴 ⊆ 𝐴 ∧ 𝐴 ⊆ ◡◡𝐴)) | |
3 | 1, 2 | bitri 267 | . 2 ⊢ (Rel 𝐴 ↔ (◡◡𝐴 ⊆ 𝐴 ∧ 𝐴 ⊆ ◡◡𝐴)) |
4 | cnvcnvss 5829 | . . 3 ⊢ ◡◡𝐴 ⊆ 𝐴 | |
5 | 4 | biantrur 526 | . 2 ⊢ (𝐴 ⊆ ◡◡𝐴 ↔ (◡◡𝐴 ⊆ 𝐴 ∧ 𝐴 ⊆ ◡◡𝐴)) |
6 | ssdif0 4171 | . 2 ⊢ (𝐴 ⊆ ◡◡𝐴 ↔ (𝐴 ∖ ◡◡𝐴) = ∅) | |
7 | 3, 5, 6 | 3bitr2i 291 | 1 ⊢ (Rel 𝐴 ↔ (𝐴 ∖ ◡◡𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 386 = wceq 1656 ∖ cdif 3795 ⊆ wss 3798 ∅c0 4144 ◡ccnv 5341 Rel wrel 5347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-br 4874 df-opab 4936 df-xp 5348 df-rel 5349 df-cnv 5350 |
This theorem is referenced by: cnvnonrel 38728 |
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