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Theorem 0qs 8780
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 8725 . 2 (∅ / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ ∅ 𝑦 = [𝑥]𝑅}
2 rex0 4354 . . 3 ¬ ∃𝑥 ∈ ∅ 𝑦 = [𝑥]𝑅
32abf 4399 . 2 {𝑦 ∣ ∃𝑥 ∈ ∅ 𝑦 = [𝑥]𝑅} = ∅
41, 3eqtri 2756 1 (∅ / 𝑅) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  {cab 2705  wrex 3066  c0 4319  [cec 8717   / cqs 8718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3058  df-rex 3067  df-dif 3948  df-nul 4320  df-qs 8725
This theorem is referenced by:  fracbas  32986
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