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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 38162 | . 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 37554 EqvRel weqvrel 37571 Disj wdisjALTV 37588 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-coss 37792 df-refrel 37893 df-cnvrefrel 37908 df-symrel 37925 df-trrel 37955 df-eqvrel 37966 df-disjALTV 38086 |
This theorem is referenced by: det0 38168 detid 38174 detidres 38176 detinidres 38177 detxrnidres 38178 |
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