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Theorem erprt 38339
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 482 . . . 4 (( Er 𝑋 ∧ (𝑥 ∈ (𝐴 / ) ∧ 𝑦 ∈ (𝐴 / ))) → Er 𝑋)
2 simprl 770 . . . 4 (( Er 𝑋 ∧ (𝑥 ∈ (𝐴 / ) ∧ 𝑦 ∈ (𝐴 / ))) → 𝑥 ∈ (𝐴 / ))
3 simprr 772 . . . 4 (( Er 𝑋 ∧ (𝑥 ∈ (𝐴 / ) ∧ 𝑦 ∈ (𝐴 / ))) → 𝑦 ∈ (𝐴 / ))
41, 2, 3qsdisj 8806 . . 3 (( Er 𝑋 ∧ (𝑥 ∈ (𝐴 / ) ∧ 𝑦 ∈ (𝐴 / ))) → (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
54ralrimivva 3196 . 2 ( Er 𝑋 → ∀𝑥 ∈ (𝐴 / )∀𝑦 ∈ (𝐴 / )(𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
6 df-prt 38338 . 2 (Prt (𝐴 / ) ↔ ∀𝑥 ∈ (𝐴 / )∀𝑦 ∈ (𝐴 / )(𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
75, 6sylibr 233 1 ( Er 𝑋 → Prt (𝐴 / ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 846   = wceq 1534  wcel 2099  wral 3057  cin 3944  c0 4318   Er wer 8715   / cqs 8717  Prt wprt 38337
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-er 8718  df-ec 8720  df-qs 8724  df-prt 38338
This theorem is referenced by: (None)
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