Theorem detlem 39254

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 39252	. 2 ⊢ ( Disj 𝑅 → EqvRel ≀ 𝑅)
2		detlem.1	. . 3 ⊢ Disj 𝑅
3	2	a1i 11	. 2 ⊢ ( EqvRel ≀ 𝑅 → Disj 𝑅)
4	1, 3	impbii 210	1 ⊢ ( Disj 𝑅 ↔ EqvRel ≀ 𝑅)

Syntax hints:   ↔ wb 207   ≀ ccoss 38551   EqvRel weqvrel 38568   Disj wdisjALTV 38587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-coss 38869  df-refrel 38960  df-cnvrefrel 38975  df-symrel 38992  df-trrel 39026  df-eqvrel 39037  df-disjALTV 39158

This theorem is referenced by:  det0  39258  detid  39264  detidres  39266  detinidres  39267  detxrnidres  39268