Theorem eqvreldisj2 39439

Description: The elements of the quotient set of an equivalence relation are disjoint (cf. eqvreldisj3 39440). (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 39438	. 2 ⊢ ( EqvRel 𝑅 → ∀𝑥 ∈ (𝐴 / 𝑅)∀𝑦 ∈ (𝐴 / 𝑅)(𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
2		dfeldisj5 39324	. 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 1563  ∀wral 3079   ∩ cin 3906  ∅c0 4288   / cqs 8681   EqvRel weqvrel 38711   ElDisj weldisj 38732

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-id 5547  df-eprel 5552  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-ec 8684  df-qs 8688  df-coss 39012  df-refrel 39103  df-cnvrefrel 39118  df-symrel 39135  df-trrel 39169  df-eqvrel 39180  df-disjALTV 39301  df-eldisj 39303

This theorem is referenced by:  eqvreldisj3  39440  eqvrelqseqdisj2  39443