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Theorem erprt 37385
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 484 . . . 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 8739 . . 3 (( Er 𝑋 ∧ (𝑥 ∈ (𝐴 / ) ∧ 𝑦 ∈ (𝐴 / ))) → (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
54ralrimivva 3194 . 2 ( Er 𝑋 → ∀𝑥 ∈ (𝐴 / )∀𝑦 ∈ (𝐴 / )(𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
6 df-prt 37384 . 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 397  wo 846   = wceq 1542  wcel 2107  wral 3061  cin 3913  c0 4286   Er wer 8651   / cqs 8653  Prt wprt 37383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-er 8654  df-ec 8656  df-qs 8660  df-prt 37384
This theorem is referenced by: (None)
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