Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
||
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 8288 | . 2 ⊢ (∅ / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ ∅ 𝑦 = [𝑥]𝑅} | |
2 | rex0 4310 | . . 3 ⊢ ¬ ∃𝑥 ∈ ∅ 𝑦 = [𝑥]𝑅 | |
3 | 2 | abf 4349 | . 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ ∅ 𝑦 = [𝑥]𝑅} = ∅ |
4 | 1, 3 | eqtri 2843 | 1 ⊢ (∅ / 𝑅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 {cab 2798 ∃wrex 3138 ∅c0 4284 [cec 8280 / cqs 8281 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-dif 3932 df-nul 4285 df-qs 8288 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |