Theorem qsdisj2 8824

Description: A quotient set is a disjoint set. (Contributed by Mario Carneiro, 10-Dec-2016.)

Assertion

Ref	Expression
qsdisj2	⊢ (𝑅 Er 𝑋 → Disj 𝑥 ∈ (𝐴 / 𝑅)𝑥)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑋   𝑥,𝑅

Proof of Theorem qsdisj2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 488	. . . 4 ⊢ ((𝑅 Er 𝑋 ∧ (𝑥 ∈ (𝐴 / 𝑅) ∧ 𝑦 ∈ (𝐴 / 𝑅))) → 𝑅 Er 𝑋)
2		simprl 783	. . . 4 ⊢ ((𝑅 Er 𝑋 ∧ (𝑥 ∈ (𝐴 / 𝑅) ∧ 𝑦 ∈ (𝐴 / 𝑅))) → 𝑥 ∈ (𝐴 / 𝑅))
3		simprr 785	. . . 4 ⊢ ((𝑅 Er 𝑋 ∧ (𝑥 ∈ (𝐴 / 𝑅) ∧ 𝑦 ∈ (𝐴 / 𝑅))) → 𝑦 ∈ (𝐴 / 𝑅))
4	1, 2, 3	qsdisj 8823	. . 3 ⊢ ((𝑅 Er 𝑋 ∧ (𝑥 ∈ (𝐴 / 𝑅) ∧ 𝑦 ∈ (𝐴 / 𝑅))) → (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
5	4	ralrimivva 3205	. 2 ⊢ (𝑅 Er 𝑋 → ∀𝑥 ∈ (𝐴 / 𝑅)∀𝑦 ∈ (𝐴 / 𝑅)(𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
6		id 23	. . 3 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
7	6	disjor 5084	. 2 ⊢ (Disj 𝑥 ∈ (𝐴 / 𝑅)𝑥 ↔ ∀𝑥 ∈ (𝐴 / 𝑅)∀𝑦 ∈ (𝐴 / 𝑅)(𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
8	5, 7	sylibr 237	1 ⊢ (𝑅 Er 𝑋 → Disj 𝑥 ∈ (𝐴 / 𝑅)𝑥)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1569   ∈ wcel 2144  ∀wral 3076   ∩ cin 4088  ∅c0 4277  Disj wdisj 5069   Er wer 8722   / cqs 8724

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-11 2193  ax-ext 2732  ax-sep 5248  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-disj 5070  df-br 5103  df-opab 5167  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-er 8725  df-ec 8727  df-qs 8731

This theorem is used by:  qshash  16003