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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 9434 | . 2 ⊢ sup(𝐵, ∅, 𝑅) = ∪ {𝑥 ∈ ∅ ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ ∅ (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} | |
2 | rab0 4375 | . . 3 ⊢ {𝑥 ∈ ∅ ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ ∅ (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} = ∅ | |
3 | 2 | unieqi 4912 | . 2 ⊢ ∪ {𝑥 ∈ ∅ ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ ∅ (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} = ∪ ∅ |
4 | uni0 4930 | . 2 ⊢ ∪ ∅ = ∅ | |
5 | 1, 3, 4 | 3eqtri 2756 | 1 ⊢ sup(𝐵, ∅, 𝑅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1533 ∀wral 3053 ∃wrex 3062 {crab 3424 ∅c0 4315 ∪ cuni 4900 class class class wbr 5139 supcsup 9432 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-in 3948 df-ss 3958 df-nul 4316 df-sn 4622 df-uni 4901 df-sup 9434 |
This theorem is referenced by: inf00 9498 |
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