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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 772 | . . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝐶)) → 𝑦𝑅𝐶) | |
6 | breq2 5151 | . . . 4 ⊢ (𝑧 = 𝐶 → (𝑦𝑅𝑧 ↔ 𝑦𝑅𝐶)) | |
7 | 6 | rspcev 3612 | . . 3 ⊢ ((𝐶 ∈ 𝐵 ∧ 𝑦𝑅𝐶) → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) |
8 | 4, 5, 7 | syl2an2r 684 | . 2 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝐶)) → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) |
9 | 1, 2, 3, 8 | eqsupd 9448 | 1 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∃wrex 3071 class class class wbr 5147 Or wor 5586 supcsup 9431 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 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 6492 df-riota 7360 df-sup 9433 |
This theorem is referenced by: suppr 9462 gsumesum 32995 supfz 34636 mblfinlem2 36464 |
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