![]() |
Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > eqvrelcoss2 | Structured version Visualization version GIF version |
Description: Two ways to express equivalent cosets. (Contributed by Peter Mazsa, 3-May-2019.) |
Ref | Expression |
---|---|
eqvrelcoss2 | ⊢ ( EqvRel ≀ 𝑅 ↔ ≀ ≀ 𝑅 ⊆ ≀ 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqvrelcoss3 37193 | . 2 ⊢ ( EqvRel ≀ 𝑅 ↔ ∀𝑥∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧)) | |
2 | cocossss 37011 | . 2 ⊢ ( ≀ ≀ 𝑅 ⊆ ≀ 𝑅 ↔ ∀𝑥∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧)) | |
3 | 1, 2 | bitr4i 277 | 1 ⊢ ( EqvRel ≀ 𝑅 ↔ ≀ ≀ 𝑅 ⊆ ≀ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1539 ⊆ wss 3935 class class class wbr 5132 ≀ ccoss 36748 EqvRel weqvrel 36765 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5283 ax-nul 5290 ax-pr 5411 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3426 df-v 3468 df-dif 3938 df-un 3940 df-in 3942 df-ss 3952 df-nul 4310 df-if 4514 df-sn 4614 df-pr 4616 df-op 4620 df-br 5133 df-opab 5195 df-id 5558 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-coss 36986 df-refrel 37087 df-symrel 37119 df-trrel 37149 df-eqvrel 37160 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |