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Theorem 0qs 8807
Description: Quotient set with the empty set. (Contributed by Peter Mazsa, 14-Sep-2019.)
Assertion
Ref Expression
0qs (∅ / 𝑅) = ∅

Proof of Theorem 0qs
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-qs 8751 . 2 (∅ / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ ∅ 𝑦 = [𝑥]𝑅}
2 rex0 4360 . . 3 ¬ ∃𝑥 ∈ ∅ 𝑦 = [𝑥]𝑅
32abf 4406 . 2 {𝑦 ∣ ∃𝑥 ∈ ∅ 𝑦 = [𝑥]𝑅} = ∅
41, 3eqtri 2765 1 (∅ / 𝑅) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  {cab 2714  wrex 3070  c0 4333  [cec 8743   / cqs 8744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-dif 3954  df-nul 4334  df-qs 8751
This theorem is referenced by:  fracbas  33307
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