Theorem 0qs 8825

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 8769	. 2 ⊢ (∅ / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ ∅ 𝑦 = [𝑥]𝑅}
2		rex0 4383	. . 3 ⊢ ¬ ∃𝑥 ∈ ∅ 𝑦 = [𝑥]𝑅
3	2	abf 4429	. 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ ∅ 𝑦 = [𝑥]𝑅} = ∅
4	1, 3	eqtri 2768	1 ⊢ (∅ / 𝑅) = ∅

Syntax hints:   = wceq 1537  {cab 2717  ∃wrex 3076  ∅c0 4352  [cec 8761   / cqs 8762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-dif 3979  df-nul 4353  df-qs 8769

This theorem is referenced by:  fracbas  33272