MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  0qs Structured version   Visualization version   GIF version

Theorem 0qs 8806
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 8750 . 2 (∅ / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ ∅ 𝑦 = [𝑥]𝑅}
2 rex0 4366 . . 3 ¬ ∃𝑥 ∈ ∅ 𝑦 = [𝑥]𝑅
32abf 4412 . 2 {𝑦 ∣ ∃𝑥 ∈ ∅ 𝑦 = [𝑥]𝑅} = ∅
41, 3eqtri 2763 1 (∅ / 𝑅) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  {cab 2712  wrex 3068  c0 4339  [cec 8742   / cqs 8743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-dif 3966  df-nul 4340  df-qs 8750
This theorem is referenced by:  fracbas  33287
  Copyright terms: Public domain W3C validator