Theorem eldisjim 39057

Description: If the elements of 𝐴 are disjoint, then it has equivalent coelements (former prter1 39174). Special case of disjim 39054. (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 39054	. 2 ⊢ ( Disj (◡ E ↾ 𝐴) → EqvRel ≀ (◡ E ↾ 𝐴))
2		df-eldisj 38962	. 2 ⊢ ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴))
3		df-coeleqvrel 38841	. 2 ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴))
4	1, 2, 3	3imtr4i 292	1 ⊢ ( ElDisj 𝐴 → CoElEqvRel 𝐴)

Syntax hints:   → wi 4   E cep 5522  ^◡ccnv 5622   ↾ cres 5625   ≀ ccoss 38353   EqvRel weqvrel 38370   CoElEqvRel wcoeleqvrel 38372   Disj wdisjALTV 38389   ElDisj weldisj 38391

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-coss 38671  df-refrel 38762  df-cnvrefrel 38777  df-symrel 38794  df-trrel 38828  df-eqvrel 38839  df-coeleqvrel 38841  df-disjALTV 38960  df-eldisj 38962

This theorem is referenced by:  mainer  39118  mainer2  39130