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| Mirrors > Home > MPE Home > Th. List > Mathboxes > symrelcoss | Structured version Visualization version GIF version | ||
| Description: The class of cosets by 𝑅 is symmetric. (Contributed by Peter Mazsa, 20-Dec-2021.) | 
| Ref | Expression | 
|---|---|
| symrelcoss | ⊢ SymRel ≀ 𝑅 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | symrelcoss2 38467 | . 2 ⊢ (◡ ≀ 𝑅 ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅) | |
| 2 | dfsymrel2 38550 | . 2 ⊢ ( SymRel ≀ 𝑅 ↔ (◡ ≀ 𝑅 ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅)) | |
| 3 | 1, 2 | mpbir 231 | 1 ⊢ SymRel ≀ 𝑅 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 ⊆ wss 3951 ◡ccnv 5684 Rel wrel 5690 ≀ ccoss 38182 SymRel wsymrel 38194 | 
| 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-ral 3062 df-rex 3071 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 df-dm 5695 df-rn 5696 df-res 5697 df-coss 38412 df-symrel 38545 | 
| This theorem is referenced by: eqvrelcoss 38618 | 
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