Theorem 0qs 8766

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.

Step	Hyp	Ref	Expression
1		df-qs 8706	. 2 ⊢ (∅ / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ ∅ 𝑦 = [𝑥]𝑅}
2		rex0 4315	. . 3 ⊢ ¬ ∃𝑥 ∈ ∅ 𝑦 = [𝑥]𝑅
3	2	abf 4371	. 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ ∅ 𝑦 = [𝑥]𝑅} = ∅
4	1, 3	eqtri 2788	1 ⊢ (∅ / 𝑅) = ∅

This proof depends on syntax axioms:   = wceq 1570  {cab 2743  ∃wrex 3091  ∅c0 4286  [cec 8698   / cqs 8699

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-dif 3909  df-nul 4287  df-qs 8706

This theorem is used by:  fracbas  33692