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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 8231 | . 2 ⊢ sup(𝐵, ∅, 𝑅) = ∪ {𝑥 ∈ ∅ ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ ∅ (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} | |
2 | rab0 3909 | . . . 4 ⊢ {𝑥 ∈ ∅ ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ ∅ (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} = ∅ | |
3 | 2 | unieqi 4381 | . . 3 ⊢ ∪ {𝑥 ∈ ∅ ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ ∅ (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} = ∪ ∅ |
4 | uni0 4401 | . . 3 ⊢ ∪ ∅ = ∅ | |
5 | 3, 4 | eqtri 2632 | . 2 ⊢ ∪ {𝑥 ∈ ∅ ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ ∅ (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} = ∅ |
6 | 1, 5 | eqtri 2632 | 1 ⊢ sup(𝐵, ∅, 𝑅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∀wral 2896 ∃wrex 2897 {crab 2900 ∅c0 3874 ∪ cuni 4372 class class class wbr 4583 supcsup 8229 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-in 3547 df-ss 3554 df-nul 3875 df-sn 4126 df-uni 4373 df-sup 8231 |
This theorem is referenced by: inf00 8294 |
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