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Theorem qsdisj2 8362
Description: A quotient set is a disjoint set. (Contributed by Mario Carneiro, 10-Dec-2016.)
Assertion
Ref Expression
qsdisj2 (𝑅 Er 𝑋Disj 𝑥 ∈ (𝐴 / 𝑅)𝑥)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑋   𝑥,𝑅

Proof of Theorem qsdisj2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 simpl 486 . . . 4 ((𝑅 Er 𝑋 ∧ (𝑥 ∈ (𝐴 / 𝑅) ∧ 𝑦 ∈ (𝐴 / 𝑅))) → 𝑅 Er 𝑋)
2 simprl 770 . . . 4 ((𝑅 Er 𝑋 ∧ (𝑥 ∈ (𝐴 / 𝑅) ∧ 𝑦 ∈ (𝐴 / 𝑅))) → 𝑥 ∈ (𝐴 / 𝑅))
3 simprr 772 . . . 4 ((𝑅 Er 𝑋 ∧ (𝑥 ∈ (𝐴 / 𝑅) ∧ 𝑦 ∈ (𝐴 / 𝑅))) → 𝑦 ∈ (𝐴 / 𝑅))
41, 2, 3qsdisj 8361 . . 3 ((𝑅 Er 𝑋 ∧ (𝑥 ∈ (𝐴 / 𝑅) ∧ 𝑦 ∈ (𝐴 / 𝑅))) → (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
54ralrimivva 3159 . 2 (𝑅 Er 𝑋 → ∀𝑥 ∈ (𝐴 / 𝑅)∀𝑦 ∈ (𝐴 / 𝑅)(𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
6 id 22 . . 3 (𝑥 = 𝑦𝑥 = 𝑦)
76disjor 5013 . 2 (Disj 𝑥 ∈ (𝐴 / 𝑅)𝑥 ↔ ∀𝑥 ∈ (𝐴 / 𝑅)∀𝑦 ∈ (𝐴 / 𝑅)(𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
85, 7sylibr 237 1 (𝑅 Er 𝑋Disj 𝑥 ∈ (𝐴 / 𝑅)𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wo 844   = wceq 1538  wcel 2112  wral 3109  cin 3883  c0 4246  Disj wdisj 4998   Er wer 8273   / cqs 8275
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-rmo 3117  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-disj 4999  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
This theorem is referenced by:  qshash  15177
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