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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 35801. (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 35694 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ≀ 𝑅𝑦 ↔ 𝑦 ≀ 𝑅𝑥)) | |
2 | 1 | el2v 3501 | . . . 4 ⊢ (𝑥 ≀ 𝑅𝑦 ↔ 𝑦 ≀ 𝑅𝑥) |
3 | 2 | biimpi 218 | . . 3 ⊢ (𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥) |
4 | 3 | gen2 1797 | . 2 ⊢ ∀𝑥∀𝑦(𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥) |
5 | relcoss 35683 | . 2 ⊢ Rel ≀ 𝑅 | |
6 | 4, 5 | pm3.2i 473 | 1 ⊢ (∀𝑥∀𝑦(𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥) ∧ Rel ≀ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1535 Vcvv 3494 class class class wbr 5066 Rel wrel 5560 ≀ ccoss 35468 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 df-coss 35674 |
This theorem is referenced by: symrelcoss2 35721 eqvrelcoss3 35868 |
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