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Theorem 0qs 35614
 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 8287 . 2 (∅ / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ ∅ 𝑦 = [𝑥]𝑅}
2 rex0 4315 . . 3 ¬ ∃𝑥 ∈ ∅ 𝑦 = [𝑥]𝑅
32abf 4354 . 2 {𝑦 ∣ ∃𝑥 ∈ ∅ 𝑦 = [𝑥]𝑅} = ∅
41, 3eqtri 2842 1 (∅ / 𝑅) = ∅
 Colors of variables: wff setvar class Syntax hints:   = wceq 1531  {cab 2797  ∃wrex 3137  ∅c0 4289  [cec 8279   / cqs 8280 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 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-dif 3937  df-nul 4290  df-qs 8287 This theorem is referenced by: (None)
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