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Mirrors > Home > MPE Home > Th. List > Mathboxes > symrelcoss2 | Structured version Visualization version GIF version |
Description: The class of cosets by 𝑅 is symmetric, see dfsymrel2 35779. (Contributed by Peter Mazsa, 27-Dec-2018.) |
Ref | Expression |
---|---|
symrelcoss2 | ⊢ (◡ ≀ 𝑅 ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | symrelcoss3 35699 | . 2 ⊢ (∀𝑥∀𝑦(𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥) ∧ Rel ≀ 𝑅) | |
2 | cnvsym 5968 | . . 3 ⊢ (◡ ≀ 𝑅 ⊆ ≀ 𝑅 ↔ ∀𝑥∀𝑦(𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥)) | |
3 | 2 | anbi1i 625 | . 2 ⊢ ((◡ ≀ 𝑅 ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅) ↔ (∀𝑥∀𝑦(𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥) ∧ Rel ≀ 𝑅)) |
4 | 1, 3 | mpbir 233 | 1 ⊢ (◡ ≀ 𝑅 ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∀wal 1531 ⊆ wss 3935 class class class wbr 5058 ◡ccnv 5548 Rel wrel 5554 ≀ ccoss 35447 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-xp 5555 df-rel 5556 df-cnv 5557 df-coss 35653 |
This theorem is referenced by: symrelcoss 35790 |
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