Theorem eldisjim 38167

Description: If the elements of 𝐴 are disjoint, then it has equivalent coelements (former prter1 38262). Special case of disjim 38164. (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 38164	. 2 ⊢ ( Disj (◡ E ↾ 𝐴) → EqvRel ≀ (◡ E ↾ 𝐴))
2		df-eldisj 38090	. 2 ⊢ ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴))
3		df-coeleqvrel 37970	. 2 ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴))
4	1, 2, 3	3imtr4i 292	1 ⊢ ( ElDisj 𝐴 → CoElEqvRel 𝐴)

Syntax hints:   → wi 4   E cep 5572  ^◡ccnv 5668   ↾ cres 5671   ≀ ccoss 37556   EqvRel weqvrel 37573   CoElEqvRel wcoeleqvrel 37575   Disj wdisjALTV 37590   ElDisj weldisj 37592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-coss 37794  df-refrel 37895  df-cnvrefrel 37910  df-symrel 37927  df-trrel 37957  df-eqvrel 37968  df-coeleqvrel 37970  df-disjALTV 38088  df-eldisj 38090

This theorem is referenced by:  mainer  38217  mainer2  38229