Theorem eqvreldisj2 39558

Description: The elements of the quotient set of an equivalence relation are disjoint (cf. eqvreldisj3 39559). (Contributed by Mario Carneiro, 10-Dec-2016.) (Revised by Peter Mazsa, 19-Sep-2021.)

Assertion

Ref	Expression
eqvreldisj2	⊢ ( EqvRel 𝑅 → ElDisj (𝐴 / 𝑅))

Proof of Theorem eqvreldisj2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqvreldisj1 39557	. 2 ⊢ ( EqvRel 𝑅 → ∀𝑥 ∈ (𝐴 / 𝑅)∀𝑦 ∈ (𝐴 / 𝑅)(𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
2		dfeldisj5 39443	. 2 ⊢ ( ElDisj (𝐴 / 𝑅) ↔ ∀𝑥 ∈ (𝐴 / 𝑅)∀𝑦 ∈ (𝐴 / 𝑅)(𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
3	1, 2	sylibr 237	1 ⊢ ( EqvRel 𝑅 → ElDisj (𝐴 / 𝑅))

Syntax hints:   → wi 4   ∨ wo 860   = wceq 1570  ∀wral 3079   ∩ cin 3905  ∅c0 4287   / cqs 8694   EqvRel weqvrel 38830   ElDisj weldisj 38851

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-id 5558  df-eprel 5563  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-ec 8697  df-qs 8701  df-coss 39131  df-refrel 39222  df-cnvrefrel 39237  df-symrel 39254  df-trrel 39288  df-eqvrel 39299  df-disjALTV 39420  df-eldisj 39422

This theorem is referenced by:  eqvreldisj3  39559  eqvrelqseqdisj2  39562