Theorem eldisjim 39199

Description: If the elements of 𝐴 are disjoint, then it has equivalent coelements (former prter1 39316). Special case of disjim 39196. (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 39196	. 2 ⊢ ( Disj (◡ E ↾ 𝐴) → EqvRel ≀ (◡ E ↾ 𝐴))
2		df-eldisj 39104	. 2 ⊢ ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴))
3		df-coeleqvrel 38983	. 2 ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴))
4	1, 2, 3	3imtr4i 292	1 ⊢ ( ElDisj 𝐴 → CoElEqvRel 𝐴)

Syntax hints:   → wi 4   E cep 5521  ^◡ccnv 5621   ↾ cres 5624   ≀ ccoss 38495   EqvRel weqvrel 38512   CoElEqvRel wcoeleqvrel 38514   Disj wdisjALTV 38531   ElDisj weldisj 38533

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 2185  ax-ext 2709  ax-sep 5231  ax-pr 5368

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-coss 38813  df-refrel 38904  df-cnvrefrel 38919  df-symrel 38936  df-trrel 38970  df-eqvrel 38981  df-coeleqvrel 38983  df-disjALTV 39102  df-eldisj 39104

This theorem is referenced by:  mainer  39260  mainer2  39272