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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 8725 | . 2 ⊢ (∅ / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ ∅ 𝑦 = [𝑥]𝑅} | |
2 | rex0 4354 | . . 3 ⊢ ¬ ∃𝑥 ∈ ∅ 𝑦 = [𝑥]𝑅 | |
3 | 2 | abf 4399 | . 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ ∅ 𝑦 = [𝑥]𝑅} = ∅ |
4 | 1, 3 | eqtri 2756 | 1 ⊢ (∅ / 𝑅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 {cab 2705 ∃wrex 3066 ∅c0 4319 [cec 8717 / cqs 8718 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3058 df-rex 3067 df-dif 3948 df-nul 4320 df-qs 8725 |
This theorem is referenced by: fracbas 32986 |
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