Theorem detid 39545

Description: The cosets by the identity relation are in equivalence relation if and only if the identity relation is disjoint. (Contributed by Peter Mazsa, 31-Dec-2021.)

Assertion

Ref	Expression
detid	⊢ ( Disj I ↔ EqvRel ≀ I )

Proof of Theorem detid

Step	Hyp	Ref	Expression
1		disjALTVid 39504	. 2 ⊢ Disj I
2	1	detlem 39535	1 ⊢ ( Disj I ↔ EqvRel ≀ I )

Syntax hints:   ↔ wb 209   I cid 5555   ≀ ccoss 38832   EqvRel weqvrel 38849   Disj wdisjALTV 38868

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 5257  ax-pr 5404

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 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-coss 39150  df-refrel 39241  df-cnvrefrel 39256  df-symrel 39273  df-trrel 39307  df-eqvrel 39318  df-disjALTV 39439

This theorem is referenced by: (None)