Theorem eldisjim 37002

Description: If the elements of 𝐴 are disjoint, then it has equivalent coelements (former prter1 37097). Special case of disjim 36999. (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 36999	. 2 ⊢ ( Disj (◡ E ↾ 𝐴) → EqvRel ≀ (◡ E ↾ 𝐴))
2		df-eldisj 36925	. 2 ⊢ ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴))
3		df-coeleqvrel 36805	. 2 ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴))
4	1, 2, 3	3imtr4i 291	1 ⊢ ( ElDisj 𝐴 → CoElEqvRel 𝐴)

Syntax hints:   → wi 4   E cep 5512  ^◡ccnv 5606   ↾ cres 5609   ≀ ccoss 36389   EqvRel weqvrel 36406   CoElEqvRel wcoeleqvrel 36408   Disj wdisjALTV 36423   ElDisj weldisj 36425

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-sep 5238  ax-nul 5245  ax-pr 5367

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3443  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4268  df-if 4472  df-sn 4572  df-pr 4574  df-op 4578  df-br 5088  df-opab 5150  df-id 5507  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-coss 36629  df-refrel 36730  df-cnvrefrel 36745  df-symrel 36762  df-trrel 36792  df-eqvrel 36803  df-coeleqvrel 36805  df-disjALTV 36923  df-eldisj 36925

This theorem is referenced by:  mainer  37052  mainer2  37064