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Mirrors > Home > MPE Home > Th. List > Mathboxes > symrelcoss3 | Structured version Visualization version GIF version |
Description: The class of cosets by 𝑅 is symmetric, see dfsymrel3 38077. (Contributed by Peter Mazsa, 28-Mar-2019.) (Revised by Peter Mazsa, 17-Sep-2021.) |
Ref | Expression |
---|---|
symrelcoss3 | ⊢ (∀𝑥∀𝑦(𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥) ∧ Rel ≀ 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brcosscnvcoss 37961 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ≀ 𝑅𝑦 ↔ 𝑦 ≀ 𝑅𝑥)) | |
2 | 1 | el2v 3471 | . . . 4 ⊢ (𝑥 ≀ 𝑅𝑦 ↔ 𝑦 ≀ 𝑅𝑥) |
3 | 2 | biimpi 215 | . . 3 ⊢ (𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥) |
4 | 3 | gen2 1790 | . 2 ⊢ ∀𝑥∀𝑦(𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥) |
5 | relcoss 37950 | . 2 ⊢ Rel ≀ 𝑅 | |
6 | 4, 5 | pm3.2i 469 | 1 ⊢ (∀𝑥∀𝑦(𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥) ∧ Rel ≀ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∀wal 1531 Vcvv 3463 class class class wbr 5143 Rel wrel 5677 ≀ ccoss 37704 |
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 3943 df-un 3945 df-ss 3957 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-coss 37938 |
This theorem is referenced by: symrelcoss2 37993 eqvrelcoss3 38145 |
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