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Mirrors > Home > MPE Home > Th. List > sup00 | Structured version Visualization version GIF version |
Description: The supremum under an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.) |
Ref | Expression |
---|---|
sup00 | ⊢ sup(𝐵, ∅, 𝑅) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-sup 8894 | . 2 ⊢ sup(𝐵, ∅, 𝑅) = ∪ {𝑥 ∈ ∅ ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ ∅ (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} | |
2 | rab0 4334 | . . 3 ⊢ {𝑥 ∈ ∅ ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ ∅ (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} = ∅ | |
3 | 2 | unieqi 4839 | . 2 ⊢ ∪ {𝑥 ∈ ∅ ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ ∅ (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} = ∪ ∅ |
4 | uni0 4857 | . 2 ⊢ ∪ ∅ = ∅ | |
5 | 1, 3, 4 | 3eqtri 2845 | 1 ⊢ sup(𝐵, ∅, 𝑅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1528 ∀wral 3135 ∃wrex 3136 {crab 3139 ∅c0 4288 ∪ cuni 4830 class class class wbr 5057 supcsup 8892 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-in 3940 df-ss 3949 df-nul 4289 df-sn 4558 df-uni 4831 df-sup 8894 |
This theorem is referenced by: inf00 8958 |
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