Theorem eldisjim 38782

Description: If the elements of 𝐴 are disjoint, then it has equivalent coelements (former prter1 38878). Special case of disjim 38779. (Contributed by Rodolfo Medina, 13-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015) (Revised by Peter Mazsa, 8-Feb-2018.) ( Revised by Peter Mazsa, 23-Sep-2021.)

Assertion

Ref	Expression
eldisjim	⊢ ( ElDisj 𝐴 → CoElEqvRel 𝐴)

Proof of Theorem eldisjim

Step	Hyp	Ref	Expression
1		disjim 38779	. 2 ⊢ ( Disj (◡ E ↾ 𝐴) → EqvRel ≀ (◡ E ↾ 𝐴))
2		df-eldisj 38705	. 2 ⊢ ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴))
3		df-coeleqvrel 38584	. 2 ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴))
4	1, 2, 3	3imtr4i 292	1 ⊢ ( ElDisj 𝐴 → CoElEqvRel 𝐴)

Syntax hints:   → wi 4   E cep 5518  ^◡ccnv 5618   ↾ cres 5621   ≀ ccoss 38175   EqvRel weqvrel 38192   CoElEqvRel wcoeleqvrel 38194   Disj wdisjALTV 38209   ElDisj weldisj 38211

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5093  df-opab 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-coss 38408  df-refrel 38509  df-cnvrefrel 38524  df-symrel 38541  df-trrel 38571  df-eqvrel 38582  df-coeleqvrel 38584  df-disjALTV 38703  df-eldisj 38705

This theorem is referenced by:  mainer  38832  mainer2  38844