Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  0qs Structured version   Visualization version   GIF version

Theorem 0qs 34582
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 7957 . 2 (∅ / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ ∅ 𝑦 = [𝑥]𝑅}
2 rex0 4104 . . 3 ¬ ∃𝑥 ∈ ∅ 𝑦 = [𝑥]𝑅
32abf 4142 . 2 {𝑦 ∣ ∃𝑥 ∈ ∅ 𝑦 = [𝑥]𝑅} = ∅
41, 3eqtri 2787 1 (∅ / 𝑅) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1652  {cab 2751  wrex 3056  c0 4081  [cec 7949   / cqs 7950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-v 3352  df-dif 3737  df-nul 4082  df-qs 7957
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator