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Mirrors > Home > MPE Home > Th. List > supmax | Structured version Visualization version GIF version |
Description: The greatest element of a set is its supremum. Note that the converse is not true; the supremum might not be an element of the set considered. (Contributed by Jeff Hoffman, 17-Jun-2008.) (Proof shortened by OpenAI, 30-Mar-2020.) |
Ref | Expression |
---|---|
supmax.1 | ⊢ (𝜑 → 𝑅 Or 𝐴) |
supmax.2 | ⊢ (𝜑 → 𝐶 ∈ 𝐴) |
supmax.3 | ⊢ (𝜑 → 𝐶 ∈ 𝐵) |
supmax.4 | ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝐶𝑅𝑦) |
Ref | Expression |
---|---|
supmax | ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | supmax.1 | . 2 ⊢ (𝜑 → 𝑅 Or 𝐴) | |
2 | supmax.2 | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝐴) | |
3 | supmax.4 | . 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝐶𝑅𝑦) | |
4 | supmax.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐵) | |
5 | simprr 769 | . . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝐶)) → 𝑦𝑅𝐶) | |
6 | breq2 5151 | . . . 4 ⊢ (𝑧 = 𝐶 → (𝑦𝑅𝑧 ↔ 𝑦𝑅𝐶)) | |
7 | 6 | rspcev 3611 | . . 3 ⊢ ((𝐶 ∈ 𝐵 ∧ 𝑦𝑅𝐶) → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) |
8 | 4, 5, 7 | syl2an2r 681 | . 2 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝐶)) → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) |
9 | 1, 2, 3, 8 | eqsupd 9454 | 1 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ∃wrex 3068 class class class wbr 5147 Or wor 5586 supcsup 9437 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-po 5587 df-so 5588 df-iota 6494 df-riota 7367 df-sup 9439 |
This theorem is referenced by: suppr 9468 gsumesum 33355 supfz 35002 mblfinlem2 36829 |
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