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Mirrors > Home > MPE Home > Th. List > sup0riota | Structured version Visualization version GIF version |
Description: The supremum of an empty set is the smallest element of the base set. (Contributed by AV, 4-Sep-2020.) |
Ref | Expression |
---|---|
sup0riota | ⊢ (𝑅 Or 𝐴 → sup(∅, 𝐴, 𝑅) = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . 3 ⊢ (𝑅 Or 𝐴 → 𝑅 Or 𝐴) | |
2 | 1 | supval2 8244 | . 2 ⊢ (𝑅 Or 𝐴 → sup(∅, 𝐴, 𝑅) = (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ ∅ ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧)))) |
3 | ral0 4028 | . . . . . 6 ⊢ ∀𝑦 ∈ ∅ ¬ 𝑥𝑅𝑦 | |
4 | 3 | biantrur 526 | . . . . 5 ⊢ (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧) ↔ (∀𝑦 ∈ ∅ ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧))) |
5 | rex0 3894 | . . . . . . 7 ⊢ ¬ ∃𝑧 ∈ ∅ 𝑦𝑅𝑧 | |
6 | imnot 354 | . . . . . . 7 ⊢ (¬ ∃𝑧 ∈ ∅ 𝑦𝑅𝑧 → ((𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧) ↔ ¬ 𝑦𝑅𝑥)) | |
7 | 5, 6 | ax-mp 5 | . . . . . 6 ⊢ ((𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧) ↔ ¬ 𝑦𝑅𝑥) |
8 | 7 | ralbii 2963 | . . . . 5 ⊢ (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧) ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥) |
9 | 4, 8 | bitr3i 265 | . . . 4 ⊢ ((∀𝑦 ∈ ∅ ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧)) ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥) |
10 | 9 | a1i 11 | . . 3 ⊢ (𝑅 Or 𝐴 → ((∀𝑦 ∈ ∅ ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧)) ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥)) |
11 | 10 | riotabidv 6513 | . 2 ⊢ (𝑅 Or 𝐴 → (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ ∅ ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧))) = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥)) |
12 | 2, 11 | eqtrd 2644 | 1 ⊢ (𝑅 Or 𝐴 → sup(∅, 𝐴, 𝑅) = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∀wral 2896 ∃wrex 2897 ∅c0 3874 class class class wbr 4583 Or wor 4958 ℩crio 6510 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-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-po 4959 df-so 4960 df-iota 5768 df-riota 6511 df-sup 8231 |
This theorem is referenced by: sup0 8255 |
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