Theorem 0qs 8791

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 8731	. 2 ⊢ (∅ / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ ∅ 𝑦 = [𝑥]𝑅}
2		rex0 4307	. . 3 ⊢ ¬ ∃𝑥 ∈ ∅ 𝑦 = [𝑥]𝑅
3	2	abf 4363	. 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ ∅ 𝑦 = [𝑥]𝑅} = ∅
4	1, 3	eqtri 2783	1 ⊢ (∅ / 𝑅) = ∅

This proof depends on syntax axioms:   = wceq 1569  {cab 2738  ∃wrex 3086  ∅c0 4277  [cec 8723   / cqs 8724

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-dif 4046  df-nul 4278  df-qs 8731

This theorem is used by:  fracbas  33917