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Mirrors > Home > MPE Home > Th. List > Mathboxes > detlem | Structured version Visualization version GIF version |
Description: If a relation is disjoint, then it is equivalent to the equivalent cosets of the relation, inference version. (Contributed by Peter Mazsa, 30-Sep-2021.) |
Ref | Expression |
---|---|
detlem.1 | ⊢ Disj 𝑅 |
Ref | Expression |
---|---|
detlem | ⊢ ( Disj 𝑅 ↔ EqvRel ≀ 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disjim 38253 | . 2 ⊢ ( Disj 𝑅 → EqvRel ≀ 𝑅) | |
2 | detlem.1 | . . 3 ⊢ Disj 𝑅 | |
3 | 2 | a1i 11 | . 2 ⊢ ( EqvRel ≀ 𝑅 → Disj 𝑅) |
4 | 1, 3 | impbii 208 | 1 ⊢ ( Disj 𝑅 ↔ EqvRel ≀ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ≀ ccoss 37648 EqvRel weqvrel 37665 Disj wdisjALTV 37682 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-opab 5211 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-coss 37883 df-refrel 37984 df-cnvrefrel 37999 df-symrel 38016 df-trrel 38046 df-eqvrel 38057 df-disjALTV 38177 |
This theorem is referenced by: det0 38259 detid 38265 detidres 38267 detinidres 38268 detxrnidres 38269 |
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