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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 37058. (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 36942 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ≀ 𝑅𝑦 ↔ 𝑦 ≀ 𝑅𝑥)) | |
2 | 1 | el2v 3452 | . . . 4 ⊢ (𝑥 ≀ 𝑅𝑦 ↔ 𝑦 ≀ 𝑅𝑥) |
3 | 2 | biimpi 215 | . . 3 ⊢ (𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥) |
4 | 3 | gen2 1799 | . 2 ⊢ ∀𝑥∀𝑦(𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥) |
5 | relcoss 36931 | . 2 ⊢ Rel ≀ 𝑅 | |
6 | 4, 5 | pm3.2i 472 | 1 ⊢ (∀𝑥∀𝑦(𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥) ∧ Rel ≀ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∀wal 1540 Vcvv 3444 class class class wbr 5106 Rel wrel 5639 ≀ ccoss 36680 |
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 2704 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 2711 df-cleq 2725 df-clel 2811 df-rab 3407 df-v 3446 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-coss 36919 |
This theorem is referenced by: symrelcoss2 36974 eqvrelcoss3 37126 |
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