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Theorem erprt 36162
 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 486 . . . 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 8361 . . 3 (( Er 𝑋 ∧ (𝑥 ∈ (𝐴 / ) ∧ 𝑦 ∈ (𝐴 / ))) → (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
54ralrimivva 3159 . 2 ( Er 𝑋 → ∀𝑥 ∈ (𝐴 / )∀𝑦 ∈ (𝐴 / )(𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
6 df-prt 36161 . 2 (Prt (𝐴 / ) ↔ ∀𝑥 ∈ (𝐴 / )∀𝑦 ∈ (𝐴 / )(𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
75, 6sylibr 237 1 ( Er 𝑋 → Prt (𝐴 / ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2112  ∀wral 3109   ∩ cin 3883  ∅c0 4246   Er wer 8273   / cqs 8275  Prt wprt 36160 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-er 8276  df-ec 8278  df-qs 8282  df-prt 36161 This theorem is referenced by: (None)
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