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Theorem erprt 33176
 Description: The quotient set of an equivalence relation is a partition. (Contributed by Rodolfo Medina, 13-Oct-2010.)
Assertion
Ref Expression
erprt ( Er 𝑋 → Prt (𝐴 / ))

Proof of Theorem erprt
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 472 . . . 4 (( Er 𝑋 ∧ (𝑥 ∈ (𝐴 / ) ∧ 𝑦 ∈ (𝐴 / ))) → Er 𝑋)
2 simprl 790 . . . 4 (( Er 𝑋 ∧ (𝑥 ∈ (𝐴 / ) ∧ 𝑦 ∈ (𝐴 / ))) → 𝑥 ∈ (𝐴 / ))
3 simprr 792 . . . 4 (( Er 𝑋 ∧ (𝑥 ∈ (𝐴 / ) ∧ 𝑦 ∈ (𝐴 / ))) → 𝑦 ∈ (𝐴 / ))
41, 2, 3qsdisj 7711 . . 3 (( Er 𝑋 ∧ (𝑥 ∈ (𝐴 / ) ∧ 𝑦 ∈ (𝐴 / ))) → (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
54ralrimivva 2954 . 2 ( Er 𝑋 → ∀𝑥 ∈ (𝐴 / )∀𝑦 ∈ (𝐴 / )(𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
6 df-prt 33175 . 2 (Prt (𝐴 / ) ↔ ∀𝑥 ∈ (𝐴 / )∀𝑦 ∈ (𝐴 / )(𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
75, 6sylibr 223 1 ( Er 𝑋 → Prt (𝐴 / ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∩ cin 3539  ∅c0 3874   Er wer 7626   / cqs 7628  Prt wprt 33174 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-er 7629  df-ec 7631  df-qs 7635  df-prt 33175 This theorem is referenced by: (None)
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