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Theorem qsdisj2 7943
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 474 . . . 4 ((𝑅 Er 𝑋 ∧ (𝑥 ∈ (𝐴 / 𝑅) ∧ 𝑦 ∈ (𝐴 / 𝑅))) → 𝑅 Er 𝑋)
2 simprl 811 . . . 4 ((𝑅 Er 𝑋 ∧ (𝑥 ∈ (𝐴 / 𝑅) ∧ 𝑦 ∈ (𝐴 / 𝑅))) → 𝑥 ∈ (𝐴 / 𝑅))
3 simprr 813 . . . 4 ((𝑅 Er 𝑋 ∧ (𝑥 ∈ (𝐴 / 𝑅) ∧ 𝑦 ∈ (𝐴 / 𝑅))) → 𝑦 ∈ (𝐴 / 𝑅))
41, 2, 3qsdisj 7942 . . 3 ((𝑅 Er 𝑋 ∧ (𝑥 ∈ (𝐴 / 𝑅) ∧ 𝑦 ∈ (𝐴 / 𝑅))) → (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
54ralrimivva 3073 . 2 (𝑅 Er 𝑋 → ∀𝑥 ∈ (𝐴 / 𝑅)∀𝑦 ∈ (𝐴 / 𝑅)(𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
6 id 22 . . 3 (𝑥 = 𝑦𝑥 = 𝑦)
76disjor 4742 . 2 (Disj 𝑥 ∈ (𝐴 / 𝑅)𝑥 ↔ ∀𝑥 ∈ (𝐴 / 𝑅)∀𝑦 ∈ (𝐴 / 𝑅)(𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
85, 7sylibr 224 1 (𝑅 Er 𝑋Disj 𝑥 ∈ (𝐴 / 𝑅)𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382  wa 383   = wceq 1596  wcel 2103  wral 3014  cin 3679  c0 4023  Disj wdisj 4728   Er wer 7859   / cqs 7861
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-rmo 3022  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-disj 4729  df-br 4761  df-opab 4821  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-er 7862  df-ec 7864  df-qs 7868
This theorem is referenced by:  qshash  14679
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