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Theorem erprt 38237
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 768 . . . 4 (( Er 𝑋 ∧ (𝑥 ∈ (𝐴 / ) ∧ 𝑦 ∈ (𝐴 / ))) → 𝑥 ∈ (𝐴 / ))
3 simprr 770 . . . 4 (( Er 𝑋 ∧ (𝑥 ∈ (𝐴 / ) ∧ 𝑦 ∈ (𝐴 / ))) → 𝑦 ∈ (𝐴 / ))
41, 2, 3qsdisj 8785 . . 3 (( Er 𝑋 ∧ (𝑥 ∈ (𝐴 / ) ∧ 𝑦 ∈ (𝐴 / ))) → (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
54ralrimivva 3192 . 2 ( Er 𝑋 → ∀𝑥 ∈ (𝐴 / )∀𝑦 ∈ (𝐴 / )(𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
6 df-prt 38236 . 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 844   = wceq 1533  wcel 2098  wral 3053  cin 3940  c0 4315   Er wer 8697   / cqs 8699  Prt wprt 38235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-er 8700  df-ec 8702  df-qs 8706  df-prt 38236
This theorem is referenced by: (None)
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