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| Description: Quotient set with the empty set. (Contributed by Peter Mazsa, 14-Sep-2019.) | 
| Ref | Expression | 
|---|---|
| 0qs | ⊢ (∅ / 𝑅) = ∅ | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-qs 8751 | . 2 ⊢ (∅ / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ ∅ 𝑦 = [𝑥]𝑅} | |
| 2 | rex0 4360 | . . 3 ⊢ ¬ ∃𝑥 ∈ ∅ 𝑦 = [𝑥]𝑅 | |
| 3 | 2 | abf 4406 | . 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ ∅ 𝑦 = [𝑥]𝑅} = ∅ | 
| 4 | 1, 3 | eqtri 2765 | 1 ⊢ (∅ / 𝑅) = ∅ | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 {cab 2714 ∃wrex 3070 ∅c0 4333 [cec 8743 / cqs 8744 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-dif 3954 df-nul 4334 df-qs 8751 | 
| This theorem is referenced by: fracbas 33307 | 
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