Theorem 0qs 8700

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 8641	. 2 ⊢ (∅ / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ ∅ 𝑦 = [𝑥]𝑅}
2		rex0 4312	. . 3 ⊢ ¬ ∃𝑥 ∈ ∅ 𝑦 = [𝑥]𝑅
3	2	abf 4358	. 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ ∅ 𝑦 = [𝑥]𝑅} = ∅
4	1, 3	eqtri 2759	1 ⊢ (∅ / 𝑅) = ∅

Syntax hints:   = wceq 1541  {cab 2714  ∃wrex 3060  ∅c0 4285  [cec 8633   / cqs 8634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-dif 3904  df-nul 4286  df-qs 8641

This theorem is referenced by:  fracbas  33387