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Mathbox for Peter Mazsa |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 8278 | . 2 ⊢ (∅ / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ ∅ 𝑦 = [𝑥]𝑅} | |
2 | rex0 4271 | . . 3 ⊢ ¬ ∃𝑥 ∈ ∅ 𝑦 = [𝑥]𝑅 | |
3 | 2 | abf 4310 | . 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ ∅ 𝑦 = [𝑥]𝑅} = ∅ |
4 | 1, 3 | eqtri 2821 | 1 ⊢ (∅ / 𝑅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 {cab 2776 ∃wrex 3107 ∅c0 4243 [cec 8270 / cqs 8271 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ral 3111 df-rex 3112 df-dif 3884 df-nul 4244 df-qs 8278 |
This theorem is referenced by: (None) |
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