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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 6142 | . . 3 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴) | |
2 | eqss 3960 | . . 3 ⊢ (◡◡𝐴 = 𝐴 ↔ (◡◡𝐴 ⊆ 𝐴 ∧ 𝐴 ⊆ ◡◡𝐴)) | |
3 | 1, 2 | bitri 275 | . 2 ⊢ (Rel 𝐴 ↔ (◡◡𝐴 ⊆ 𝐴 ∧ 𝐴 ⊆ ◡◡𝐴)) |
4 | cnvcnvss 6147 | . . 3 ⊢ ◡◡𝐴 ⊆ 𝐴 | |
5 | 4 | biantrur 532 | . 2 ⊢ (𝐴 ⊆ ◡◡𝐴 ↔ (◡◡𝐴 ⊆ 𝐴 ∧ 𝐴 ⊆ ◡◡𝐴)) |
6 | ssdif0 4324 | . 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 397 = wceq 1542 ∖ cdif 3908 ⊆ wss 3911 ∅c0 4283 ◡ccnv 5633 Rel wrel 5639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-xp 5640 df-rel 5641 df-cnv 5642 |
This theorem is referenced by: cnvnonrel 41867 |
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