Theorem detlem 39516

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.)

Hypothesis

Ref	Expression
detlem.1	⊢ Disj 𝑅

Assertion

Ref	Expression
detlem	⊢ ( Disj 𝑅 ↔ EqvRel ≀ 𝑅)

Proof of Theorem detlem

Step	Hyp	Ref	Expression
1		disjim 39514	. 2 ⊢ ( Disj 𝑅 → EqvRel ≀ 𝑅)
2		detlem.1	. . 3 ⊢ Disj 𝑅
3	2	a1i 11	. 2 ⊢ ( EqvRel ≀ 𝑅 → Disj 𝑅)
4	1, 3	impbii 212	1 ⊢ ( Disj 𝑅 ↔ EqvRel ≀ 𝑅)

Syntax hints:   ↔ wb 209   ≀ ccoss 38813   EqvRel weqvrel 38830   Disj wdisjALTV 38849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-coss 39131  df-refrel 39222  df-cnvrefrel 39237  df-symrel 39254  df-trrel 39288  df-eqvrel 39299  df-disjALTV 39420

This theorem is referenced by:  det0  39520  detid  39526  detidres  39528  detinidres  39529  detxrnidres  39530