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Mirrors > Home > MPE Home > Th. List > 0qs | Structured version Visualization version GIF version |
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 8750 | . 2 ⊢ (∅ / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ ∅ 𝑦 = [𝑥]𝑅} | |
2 | rex0 4366 | . . 3 ⊢ ¬ ∃𝑥 ∈ ∅ 𝑦 = [𝑥]𝑅 | |
3 | 2 | abf 4412 | . 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ ∅ 𝑦 = [𝑥]𝑅} = ∅ |
4 | 1, 3 | eqtri 2763 | 1 ⊢ (∅ / 𝑅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 {cab 2712 ∃wrex 3068 ∅c0 4339 [cec 8742 / cqs 8743 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-dif 3966 df-nul 4340 df-qs 8750 |
This theorem is referenced by: fracbas 33287 |
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